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Math 110: College Algebra
Course Objectives: This course is designed to cause the student to learn traditional college algebra concepts and problem solving skills. It should serve to prepare students for Math 180, Math 230, Math 265, or Math 270.
Prerequisite: Acceptable placement score or C grade in Math 001 or equivalent (typically high school algebra). See me right away if you have a question about your math background as it relates to this reqt. Text: College Algebra: Concepts and Models, second edition, by Larson, Hostetler, and Hodgkins. Heath. 1996. References: There are a number of college algebra texts in my office and in the library.
Note: Grades are based on points allocated above. No extra credit. Typically, 90%+ is A, 80%+ is B, 70%+ is C, and 60%+ is D.
Note: Quizzes are "open book, open notes"; exams are "closed book, closed notes".
Note: All tests taken in regular classroom at scheduled times. No exams taken in learning center unless diagnosed learning disability exists (verified by Mr. Wojeichowski in writing).
Note: Final exam must be taken at regularly scheduled time (Tuesday, December 14, 7:40 - 9:40) unless approved in writing by the Dean.
Attendance: Required. See Viterbo College catalog, page 36. All guidelines followed.
A valid verifiable excuse must be presented in order to make up missed exams or quizzes. "I overslept", "My ride is leaving early for vacation/ the weekend/ etc.", "I had a busy week and didn't have time to study" are examples of NON-valid excuses. Make-up exams for valid excused absences must be done in a timely manner, usually within one week of return.
Calculating Equipment: Hand-held calculators are permitted for quizzes and exams.
Cheating: First offense - half credit on pertinent work; second offense - zero credit on pertinent work; third offense - failure in the course.Note: accommodation for special test-taking needs will be made only after these needs are confirmed in writing by Mr. Wojciechowski.
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Math Competitions
There are 6 High School Contests each year, with 6 questions per contest. There is a 30 minute time limit for each contest. On each contest, the last two questions are generally more difficult than the first four. The final question on each contest is intended to challenge the very best mathematics students. The problems require no knowledge beyond secondary school mathematics. No knowledge of calculus is required to solve any of these problems. Two to four of the questions on each contest only require a knowledge of elementary algebra. Starting with the 1992-93 school year, students have been permitted to use any calculator on any of our contests.
** Contests are held in Roberts Auditorium in the Math and Science Building
2009-2010 High School Contest Dates
Contest #
Official Date* (Tuesdays)
HS Contest 1
October 18, 2011
HS Contest 2
November 15, 2011
HS Contest 3
December 13, 2011
HS Contest 4
January 10, 2012
HS Contest 5
February 14, 2012
HS Contest 6
March 13, 2012
*Alternate contest dates may be scheduled one week before the contest dates, in the event of scheduling conflicts.
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\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2009d.00593}
\itemau{Jiang, Zhonghong}
\itemti{The dynamic geometry software as an effective learning and teaching tool.}
\itemso{Electron. J. Math. Technol. 1, No. 3, 245-256, electronic only (2007).}
\itemab
Summary: This article describes how the use of dynamic geometry software has helped preservice teachers develop their abilities in three aspects: 1) challenging problem solving; 2) mathematical modeling; and 3) constructing student-centered teaching projects. The examples given indicate that for some of the challenging problems that are presented to students, it is almost impossible or very hard to manually make correct drawings. To overcome this difficulty, the use of dynamic geometry software seems to be critical, or at least very desirable. In addition, the use of the software can stimulate students' insight of problem solving and provide an easy and convincing way of verifying the solution. Moreover, students can construct accurate visual representations to model real world situations very efficiently by using transformations in dynamic geometry software. This can save time significantly so that students can concentrate on more conceptual oriented tasks. Good teaching projects that take advantage of dynamic geometry software can also effectively enhance school children's mathematics learning. The supplemental materials that accompany this paper can be found online at the following URL: \url{
\itemrv{~}
\itemcc{U70 U50 R20}
\itemut{}
\itemli{
\end
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The idea of powerful mathematics delivered through very visual, interactive, point-and-click methods has launched a new generation of teaching and learning techniques in mathematics. Video Demonstration: What is Clickable Math?
Maple T.A. users can take advantage of thousands of free questions on calculus, precalculus, algebra, physics, and more. Questions and assignments can be freely used, recombined, and modified. Browse all Maple T.A. Content
Precalculus
Each topic in the Precalculus classroom content includes Maple T.A. questions to test students understanding and provide extra practice.
Calculus 1
This content covers the complete first semester of an introductory honors calculus course, and provides weekly assignments. This material has been used at the University of Guelph for the last three years.
Calculus 2
This content covers the complete second semester of an introductory honors calculus course, and provides weekly assignments. This material has been used at the University of Guelph for the last three years.
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As electromagnetics, photonics, and materials science evolve, it is increasingly important for students and practitioners in the physical sciences and engineering to understand vector calculus and tensor analysis. This book provides a review of vector calculus. This review includes necessary excursions into tensor analysis intended as the reader's first exposure to tensors, making aspects of tensors understandable to advanced undergraduate students. This book will also prepare the reader for more advanced studies in vector calculus and tensor analysis.
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Essential Maths for Engineering and Construction
$21.87
Reward Money:
$0.76
Promo Eligible
Don't let your mathematical skills fail you! In Engineering, Construction, and Science examinations, marks are often lost through carelessness or from not properly understanding the mathematics involved. When there are only a few marks on offer for a part of a question, there may be full marks for a right answer and none for a wrong one, regardless of the thought that went into the answer.
If you want to avoid losing these marks by improving the clarity both of your mathematical work and your mathematical understanding, then Essential Maths for Engineering and Construction is the book for you.We all make mistakes; who doesn't? But mistakes can be avoided when we understand why we make them. Taking mistakes commonly made by undergraduate students as its entry point, this book not only looks at how you can prevent mistakes, but also provides a primer for the fundamental mathematical skills required for your degree discipline.
Whether you struggle with different types of interest rates, geometry, statistics, calculus, or any of the other mathematical areas vital to your degree, this book will guide you around the pitfalls
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Course
Philosophy and Description This course is designed for students
majoring in the mathematical and physical sciences, engineering,
or mathematics education and for students minoring in mathematics
or mathematics education. Calculus is the foundation for most of
the mathematics studied at the university level. The mastery of
calculus requires well-developed skills, clear conceptual
understanding, and the ability to model phenomena in a variety of
settings. Calculus 1 develops the concepts of limit, derivative,
and integral, and is fundamental for many fields of mathematics.
This course contributes to all the expected learning
outcomes of the Mathematics BS.
Although the primary objective of this class is to introduce you
to fundamental ideas in Calculus, you will also improve your
mathematical problem-solving and reasoning skills. I also hope
that you will develop a greater appreciation for mathematics in
general.
Course
Learning Outcomes
Students are expected to master the "core topics" of Math 112,
consisting of the material in the first five chapters of the text.
In particular, students are expected to master the following
topics:
Limits and
Continuity: Students will be able to work with limits
and continuity offunctions. In
particular, they will:
compute limits of functions described
algebraically and graphically,
determine when functions are
continuous, identify discontinuities and asymptotes,
use definitions, limit laws, or the
Squeeze theorem to prove a limit is a given value,
state and apply the Intermediate Value
Theorem to prove facts about continuous functions.
The Derivative and Applications: Students will
be able to take and apply derivativesfluently. In particular, they will:
state the definition of a derivative,
use it to compute derivatives, and explain its interpretations as slope and rate of
change,
compute derivatives of elementary
functions and their combinations using derivative rules,
prove facts about derivatives using
the definition, derivative rules, as well as Rolle's theorem and the Mean Value Theorem,
use the derivative to sketch curves,
optimize functions, evaluate limits with L'Hˆopital's rule, find related rates, and approximate
functions and their roots.
The Integral: Students will be able to
integrate simple functions, and know when and how to apply the integral. In particular, they will:
state the definition of the definite
integral, approximate it by Riemann sums, compute it as a limit of Riemann sums, and
explain its interpretations as area and net change,
use properties of the integral, such
as those pertaining to the interval of integration, to simplify, estimate, and compute
integrals,
state and apply the Fundamental
Theorem of Calculus, both to evaluate integrals, and to find derivatives of functions
defined as definite integrals,
use simple substitutions to evaluate
integrals.
Prerequisites Math 110 and 111 or the equivalent. This includes College
Algebra and Trigonometry, but could also be satisfied with a
Precalculus course in Precalculus. You are also required to take a
pretest in order to
exhibit competency in these areas (see Description of Activities).
Preparation Time
Adequately prepared students should expect to spend a minimum of
three hours of work for each credit hour. This means you should
expect to spend a minimum of 12 hours per week for Math 112. A
minimal time commitment typically leads to an average grade B-/C+
or lower. Much more time may be required to achieve excellence.
You will need to have access to a computer with Internet access. I
also recommend that you acquire a graphing calculator, but this is
not required. If you already have a graphing calculator I suggest
you use it. If you are considering purchasing one I suggest the
TI-89 because that is the one I will typically use for demonstration
purposes in class.
There are excellent, FREE tutors available in the Math lab (159 TMCB)
and clearly marked areas where you can get Math 112 tutoring. You'll
also find there a Private
Tutor list--individuals who you can hire for one-on-one
tutoring.
Course Content There are a number of activities we will do this semester in
order to accomplish the course objective. Your grade for the course
will be determined by your performance in these activities,
including your attendance, preparation and participation. Details on
each course activity, including the number of points they are worth
and a rubric for how they will be graded, can be found on the Description of Activities page.
The Course Outline details
when these activities will take place throughout the semester.
Late work will not be
graded. If you need to be absent, turn in your
work before you leave or have a friend, roomate or classmate turn it
in for you.
Your final grade will be determined based on the
following standard scale:
Grade
%
Grade
%
A
93-100C+
77-79
E
< 60
BYU Honor Code
In keeping with the principles of
the BYU
Honor Code, students are expected to be honest in all of
their academic work. Academic honesty means, most
fundamentally, that any work you present as your own must in
fact be your own work and not that of another. Violations of
this principle may result in a failing grade in the course and
additional disciplinary action by the university. Students are
also expected to adhere to the Dress and Grooming Standards.
Adherence demonstrates respect for yourself and others and
ensures an effective learning and working environment. It is
the university's expectation, and my own expectation in class,
that each student will abide by all Honor Code standards.
Please call the Honor Code Office at 422-2847 if you have
questions about those standards.
The first
injunction of the BYU Honor Code is the call to be honest.
Students come to the university not only to improve their
minds, gain knowledge, and develop skills that will assist
them in their life's work, but also to build character.
President David O. McKay taught that 'character is the highest
aim of education' (The
Aims of a BYU Education, p. 6). It is the purpose of the
BYU Academic Honesty Policy to assist in fulfilling that aim.
BYU students should seek to be totally honest in their
dealings with others. They should complete their own work and
be evaluated based upon that work. They should avoid academic
dishonesty and misconduct in all its forms, including but not
limited to plagiarism, fabrication or falsification, cheating,
and other academic misconduct.
Preventing Sexual Discrimination and
Harassment
Title IX of the
Education Amendments of 1972 prohibits sex discrimination
against any participant in an educational program or activity
that receives federal funds. The act is intended to eliminate
sex discrimination in education. Title IX covers
discrimination in programs, admissions, activities, and
student-to-student sexual harassment. BYU's policy against
sexual harassment extends not only to employees of the
university, but to students as well. If you encounter unlawful
sexual harassment or gender-based discrimination, please talk
to your professor; contact the Equal Employment Office at
422-5895 or 367-5689 (24-hours); or contact the Honor Code
Office at 422-2847.
Students with Disabilities
Brigham Young University is
committed to providing a working and learning atmosphere that
reasonably accommodates qualified persons with disabilities.
If you have any disability which may impair your ability to
complete this course successfully, please contact the Services
for Students with Disabilities Office (422-2767). Reasonable
academic accommodations are reviewed for all students who have
qualified, documented disabilities. Services are coordinated
with the student and instructor by the SSD Office. If you need
assistance or if you feel you have been unlawfully
discriminated against on the basis of disability, you may seek
resolution through established grievance policy and procedures
by contacting the Equal Employment Office at 422-5895, D-285
ASB. For more information go to the University
Accessibility Center web site. Check out the UAC
Volunteer Website if you are interested in mentoring a
student with a disability.
|
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more.
Course Hero has millions of course specific materials providing students with the best way to expand
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Using Movies as a Means of Literary Analysis Presented by Lynn Knowles North Star of Texas Writing Project June 7, 2004 knowleslm@lisd.net Quick WriteWhat is your favorite movie? Brainstorm & Discuss Why do we watch movies? What
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AQA GCSE Mathematics (Linear)
Written by experienced teachers and subject experts, our books and online resources for Foundation and Higher tier will provide all the support you need to deliver the linear AQA GCSE Mathematics specification.
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Math 370
ALGEBRA
This is the first half of a year long course in algebra,
with emphasis on the ways in which algebra and geometry interact and
complement each other. We
will cover some abstract linear algebra (vector spaces and linear
transformations), groups and group actions and symmetry.
This corresponds more or
less to the following sections
from Artin's book: 1.1--1.5, 2.1--2.10, 3.1--3.6,
4.1--4.6, 5.1--5.3, 5.5--5.8, 6.1--6.4, 6.6--6.8.
A basic goal of the course is to use the abstract theory to develop
an intuition in concrete examples and learn to understand
and produce sound mathematical arguments.
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: College Algebra. 0-5-5. Preq., Mathematics ACT score between 18 and 21 inclusive, or Mathematics SAT score between 430 and 510 inclusive, or Placement by Exam, or MATH 099. This course covers the same material as MATH 101 with supplementary material including rational exponents, integer exponents, multiplying polynomials, factoring, rational expressions. Credit will not be given for both MATH 100 and MATH 101.
111: Precalculus Algebra. 0-3-3. Preq., Mathematics ACT score is greater than or equal to 26, or Placement by Exam, or MATH 100 or, or Mathematics SAT score is greater than or equal to 590, or Placement by Exam, or MATH 100 or114: Survey of Mathematics. 0-3-3. Preq., Mathematics ACT score is greater than or equal to 26, or Mathematics SAT score is greater than or equal to 590, or Placement by Exam, or MATH 100 or 101. Logic, counting principles, probability and statistics, systems of equations, geometry, mathematics of finance, nature of graphs. For liberal arts degree programs.
125: Algebra for Management and Social Sciences. 0-3-3. Preq., Mathematics ACT score is greater than or equal to 26, or Mathematics SAT score is greater than or equal to 590, or Placement by Exam, or MATH 100 or 101. Linear and quadratic equations and functions, graphs, matrices, systems of linear equations, mathematics of finance, sets, probability and statistics, exponential and logarithmic functions.
203: Introduction to Number Structure. 0-3-3. Preq., Mathematics ACT score is greater than or equal to 26, or Mathematics SAT score is greater than or equal to 590, or Placement by Exam, or MATH 100 or 101. Developing number sense and concepts underlying computation, estimation, pattern recognition, and function definition. Studying number relationships, systems, and theory. Applying algebraic concepts to solve problems is greater than or equal to 26, or Mathematics SAT score is greater than or equal to 590, or Placement by Exam, or MATH 100 or240: Engineering Mathematics I. 3-2-3. Preq., Mathematics ACT score is greater than or equal to 26, or Mathematics SAT score is greater than or equal to 590, or Placement by Exam, or MATH 100 or 101. Coreq., ENGR 120 and CHEM 100. Functions, graphs, polynomial functions; exponential and logarithmic functions and equations; introduction to analytic geometry; limits; derivatives; continuity; and some application of differentiation. Credit will not be given for MATH 240 if credit is given for MATH 111 or 220 or 222 or 230.
241: Engineering Mathematics II. 0-3-3. Preq., MATH 240. Coreq., ENGR 121 and CHEM 101. Differentiation rules; trigonometric reduction formulas, functions, graphs, inverse functions, equations; derivatives of algebraic, exponential, logarithmic, and trigonometric functions; some application of differentiation. . Credit will not be given for MATH 241 if credit is given for MATH 111 or 112 or 220 or 222 or 230.
242: Engineering Mathematics III. 0-3-3. Preq., MATH 241. Coreq., ENGR 122 and PHYS 201. Applications of differentiation; analytic geometry; antidifferentiation; techniques of integration; and selected topics. Credit will not be given for MATH 242 if credit is given for MATH 220 or 222 or 231.
243: Engineering Mathematics IV. 0-3-3. Preq., MATH 242. Coreq., ENGR 220 and MEMT 201. Improper integrals, single variable continuous statistics, vectors, three-dimensional coordinates, differentiation of functions of several variables, introduction to multivariate integration. Credit will not be given for MATH 243 if credit is given for MATH 232.
245: Engineering Mathematics VI. 0-3-3. Preq., MATH 244. Coreq., ENGR 222. Power series, Taylor series, use of series to solve differential equations, LaPlace transforms, systems of ordinary differential equations. Credit will not be given for MATH 245 if credit is given for MATH 350
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Book Description: This book is for students following a module in numerical methods, numerical techniques, or numerical analysis. It approaches the subject from a pragmatic viewpoint, appropriate for the modern student. The theory is kept to a minimum commensurate with comprehensive coverage of the subject and it contains abundant worked examples which provide easy understanding through a clear and concise theoretical treatment.
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up-to-date, broad scope textbook suitable for undergraduates starting on computational physics courses. It shows how to use computers to solve mathematical problems in physics and teaches a variety of numerical approaches. It includes exercises, examples of programs and online resources at
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Category Archives: Learning Tools
The school term has just started and it was exciting to meet familiar faces and new ones in our weekly tuition classes for both O-Level Maths, O-Level Chemistry and A-Level H2 Chemistry.
While I was teaching in my Secondary 3 Additional Mathematics class on the usage of their Scientific Calculator to solve linear simultaneous equations, one student pointed out there was a new calculator which is much more powerful in terms of functions. Another student in the O-Level class mentioned it too. I was excited about this new calculator!
I went searching for calculators in our centre and managed to find one in the cupboard! Casio fx-95ES Plus. It looks exactly like my current Casio model except it has additional functions like:
Displaying answers in surd forms! (Useful for Trigonometry, Surd!
Solving quadratic inequality! (Many students can't seem to get this right!)
However, when I went to the Ministry of Education Approved Calculator List, Casio fx-95ES Plus wasn't in that list. Since this calculator was bought by my colleague Sean Chua from Popular bookstore last November, I was thinking maybe MOE has yet to update the list so I emailed them to confirm. And attached was their reply.
Lesson learnt:
Not all calculators sold in Popular bookstores are approved for usage in major examinations.
We must still learn our concepts well instead of relying only on calculator to help us!
Well, I'm going into the topic of Quadratic Inequality in a few weeks time, join me in my weekly O-Level A-Maths tuition classes before it's too late! Go to NOW for the details!
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As a community service for those unable to purchase the
book, it is available for free reading in its entirety on
Google Books, but
is not downloadable. If you are benefitting from the free version, please consider
purchasing it on Amazon in Kindle or
paperback format. Please consider posting a review of it on
Amazon or elsewhere, or spreading the word about it to
others who may benefit.
The Math Made a Bit Easier Book Series
The first five books in my self-published Math Made a Bit
Easier book series are now available for purchase exclusively on Amazon.com.
For more information, please see below or
contact me with any
questions about the books. You can use the "Look Inside" and
"Search Inside" functionality on Amazon.
You can also read the books for free in their entirety via Google
Books, but they are not downloadable. If you are
benefitting from the free versions of the books, please consider
purchasing them on Amazon, or writing reviews of them on
Amazon, or at least spreading the word about the books to others who
may benefit.
Books in the Made Made a Bit Easier Series Now
Available
Math Made a Bit Easier: Basic Math Explained in Plain English
This is the first book in the series, and should be your starting
point unless you are absolutely certain that you fully understand
all of the concepts presented. If you do not, later books
in the series will prove to be difficult and frustrating. The book
starts with a review of basic arithmetic, followed by basic
operations, negative numbers, fractions, decimals, percents, and
basic probability and statistics. This is the foundation of all
math. The space devoted to each topic is proportional to how
difficult most students find the topic, as well as how important the
topic is in preparation for later math studies. The material is
explained conversationally and "in plain English" as promised by the
book's subtitle, without talking down to the reader, and without the
use of contrived examples or cartoonish illustrations.
The book concludes with a chapter on how to effectively study
math and improve scores on exams. Like the rest of the book, the
chapter takes a unique standpoint on the matter, and offers
suggestions including how to get oneself into the proper mental
and emotional mindset for being successful with math. Available
exclusively on Amazon.com for $6.50. The book can be read for free in its
entirety on Google Books via
this link.
Basic Algebra and Geometry Made a Bit Easier: Concepts Explained in
Plain English
This is the fourth in the series and the second "main" book. It
covers basic algebra and geometry. The scope of this book should be
sufficient for the algebra and geometry components of exams such as
the GED, (P)SAT, and career-based exams. The book includes some
practice exercises and self-tests, as well as points for review.
Available exclusively on Amazon.com for $5.25. The book can be
read for free in its entirety on Google Books via
this link.
April 2011: Anticipated release of an
independent, self-published weight loss book and companion journal. It
will be extremely practical, and very different than typical books
on the matter. It is my hope that this will be the book that really
gets through to people, and is not "just another weight loss book."
The main book will be available for purchase exclusively on Amazon.com
in paperback for $6.25, with a Kindle edition (no DRM) price of $2.99.
It will be also available for free reading in its entirety on Google
Books. The companion journal will be available in paperback
format for $4.25. Details soon.
June 2011: A test prep review book
covering basic math, algebra, and geometry, intended only for the
purposes of review, and for students who have unfortunately left their studying
for the last minute, and do not have the time to absorb the 450+
pages of material from the first two main books. Estimated price $4.95
July 2011: An 8"x10" book of worksheets
aligned to the Basic Math books of the series, with license to
freely reproduce and distribute for educational purposes. Estimated price $4.95
August 2011: An 8"x10" book of worksheets
aligned to the Basic Algebra and Geometry books of the series, with
license to freely reproduce and distribute for educational purposes. Estimated price $4.95
September 2011: A book focused specifically on
fractions, decimals, and percents, all explained extremely slowly,
targeted at parents, homeschoolers, and older students. Estimated price $5.95
October 2011: A book focused on how to teach math
to young children, targeted at parents and homeschoolers. Estimated price $4.95
November 2011: The third main book in the
series will cover more advanced topics in algebra and geometry. The scope of this book should be more than sufficient for the
algebra and geometry components of exams such as the SAT and high
school exit exams. It will also cover selected topics from the GRE and
GMAT. Not every topic from the aforementioned exams will be covered. The omitted topics will include those which students
typically
have little difficulty with, as well as topics which are considered relatively minor.
The material will be covered at a very fast pace, and will focus on the
concepts themselves, as opposed to rote exercises. Estimated price $6.95
December 2011: A book of lesson plans for
teachers, tutors, parents, and homeschoolers to accompany the third
main book in the series (Topics in Algebra and Geometry). Estimated price $4.95
Early 2012: Future books on
higher level math and/or exam-specific math TBD. The
material covered in the three main books should be more than
sufficient for most standardized exams, and in most cases even the
material in just the first two main books should suffice unless you are
pursuing additional math goals.
About the Author
Larry
Zafran was born and raised in Queens, NY where he tutored and taught
math in public and private schools. He has a Bachelors Degree in
Computer Science from Queens College where he graduated with highest
honors, and has earned most of the credits toward a Masters Degree
in Secondary Math Education.
He is a dedicated student of the piano, and the leader of a large
and active group of board game players which focuses on abstract
strategy games from Europe.
He presently lives in Cary, NC where he works as an independent math
tutor, writer, and webmaster.
Ordering, Shipping, and Returns Information
All orders and returns are fully handled directly by Amazon.com.
The books are part of Amazon's Prime program which means that Amazon
Prime members can order the books with free two-day shipping, and
non-members can get free economy shipping if ordered with $25 worth
of eligible products. The books can be ordered standalone for
Amazon's regular shipping rates.
Online Support for the Books
I provide full, free online support related to my books.
Please contact me via the
Ask a Math
Question page if anything in the books is confusing for you, or
if you would like additional information or help on a topic, or if
you would like to check your understanding of a concept.
Errata for Math Made a Bit Easier
These errors are in the 1st ed. (with leaves on the
cover), but have been fixed in the revised edition
On page 89, "2-to-the-power-of-4" is incorrectly evaluated as 8
due to an editing error. It is equal to 16, making the involved sum
equal to 29 and not 21.
On page 214, the repeating decimal 0.357 incorrectly has a double
bar over it. We always use a single bar over any repeating
decimal digit(s).
Errata for Math Made a Bit Easier Workbook
These errors are in the 1st ed. (with leaves on the
cover), but have been fixed in the revised edition
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Department Philosophy - This course is designed to challenge students to meet their God given potential and to present opportunities for them to apply mathematical concepts to solve real-life problems.
MATHEMATICS DEPARTMENT
With the exception of accounting students, all students are required to have a graphing calculator either a TI 83 plus, TI 84 plus silver edition, or TI- Nspire CAS version ( preferable).
Algebra 1a (Full Year; 1 credit) Course open to: Freshman Prerequisite: Departmental Approval Course Description: This course is designed to prepare students to be able to successfully complete a full high school mathematics curriculum. Areas of emphasis will include a review of the fundamental operations of arithmetic and basic concepts of rational numbers in both fractional and decimal form. In addition, students will learn to solve first-degree equations, perform operations on polynomials, graph first-degree equations, and apply these skills to problem solving. Calculators will be used frequently. Text: SpringBoard Mathematics with Meaning Algebra 1, The College Board's official Pre-AP program, (2010).
College Prep Algebra 1b (Full Year; 1 credit) Course Open to: Freshmen, Sophomores Course Description: This course is the basis for all subsequent work in mathematics. Students will be introduced to fundamental concepts of expressions, equations, inequalities, and functions. This course introduces the student to the concepts and skills of Algebra to prepare her for future courses in Geometry and Algebra 2. Topics include operations with real numbers, solving linear equations and inequalities, factoring, graphing, and functions. Students will integrate concepts of geometry, statistics, data analysis, probability and discrete mathematics in real-world applications to connect mathematics to other subjects such as science, history and health. Text: SpringBoard Mathematics with Meaning Algebra 1, The College Board's official Pre-AP program, (2010).
Honors Algebra 1 (Full Year; 1 credit) Course Open to: Freshmen, Sophomores Course Description: This course introduces the student to the concepts and skills of Algebra to prepare her for future courses in Honors Algebra and Honors Geometry. Topics include such as solving linear and quadratic equations and inequalities, factoring, algebraic fractions, fractional equations, graphing, and functions. Topics are covered to a greater depth than in College Prep Algebra I. Text:SpringBoard Mathematics with Meaning Algebra 1, The College Board's official Pre-AP program, (2010).
Algebra II (Full Year; 1 credit) Course open to: Sophomores, Juniors Course Description: After a thorough review of the topics covered in Algebra I, especially those related to the simplification of polynomial, rational and radical expressions and the solution of polynomial, quadratic, rational, and radical equations, the course proceeds to a deeper look at the notions of functions and relations, systems of linear and quadratic equations, exponents and logarithms, complex numbers, and sequences and series. Text: SpringBoard Mathematics with Meaning Algebra II, The College Board's official Pre-AP program, (2010).
Honors Algebra II (Full Year; 1 credit) Course open to: Freshmen, Sophomores, Juniors Prerequisite: At least 85+ average in Honors Algebra I and departmental approval Course Description: Honors Algebra II is a course designed for students who have excelled in Algebra I Honors. Topics included are functions and relations, systems of linear and quadratic equations, exponents and logarithms, complex numbers, and sequences and series. Graphing calculator is required. Students will solve problems of increasing complexity using previously learned concepts of algebra.
Text: SpringBoard Mathematics with Meaning Algebra II, The College Board's official Pre-AP program, (2010).
College Prep Geometry (Full Year; 1 credit) Course open to: Sophomores, Juniors Prerequisite: Algebra I
Course Description: This course introduces the student to all the basic concepts and correlate formal geometric theorems with the student's practical experience with the world around him. Topics included in the course are the study of relations of lines, parallel lines relationships, angles, right triangles, similar triangles, surfaces, circles, construction, area, volume, and deductive proofs. Lessons focus on one objective
Text: SpringBoard Mathematics with Meaning Geometry, The College Board's official Pre-AP program, (2010).
Honors Geometry: (Full Year; 1 credit) Course open to: Sophomores, Juniors Prerequisite: At least 85+ average in Honors Algebra II and departmental approval
Description: This course examines the properties, measurement, and relation of lines, angles, surfaces, and solids. Intuitive thought and the development of logical thought are emphasized and reinforced. Topics included are deductive proof, angle relationships, parallel lines, right triangles, circles, constructions and loci, and area of volume. Deductive proof is emphasized.
Text: SpringBoard Mathematics with Meaning Geometry, The College Board's official Pre-AP program, (2010).
Advanced Math and Trigonometry (Full Year; 1 credit) Course open to: Seniors Prerequisite: Algebra 2 and departmental approval Course Description: This class is designed to complete the student's preparation for college math. This course begins with a brief review of Algebra II. Which can include topics such as applications problems, graphing, units on relations, functions, and polynomials. The complex number system, conics, exponents, logarithms, sequence and series, probability, statistics, trigonometric functions, and limits will be covered as well. Students will use the graphing calculator while applying logic, reasoning, and critical thinking skills to analyze and solve problems. SAT problems will be implemented into the lessons.
Text: SpringBoard Mathematics with Meaning Precalculus, The College Board's official Pre-AP program, (2010).
Honors Pre-Calculus (Full Year; 1 credit) Course open to: Juniors, Seniors Prerequisite: At least 85+ average in Honors Algebra 2 and departmental approval Course Description: This course is a preparation for the study of calculus. It is study of linear relations and functions, the nature of graphs, polynomial and rational functions, trigonometry, graphs and inverses of the trigonometric functions, trigonometric identities and equations, polar coordinates and complex numbers, conics, exponential and logarithmic functions, sequence and series, and limits, derivatives, and integrals. Students will use the graphing calculator while applying logic, reasoning, and critical thinking skills to analyze and solve problems.
Text: SpringBoard Mathematics with Meaning Precalculus, College Board's official Pre-AP program, (2010).
AP Calculus (Full Year; 1 credit) Course open to: Seniors Prerequisite: At least 85+ average in Honors Pre-Calculus and departmental approval. Course Description: AP Calculus class is for students who succeed in algebra, trigonometry, geometry, and precalculus. Students must take AP exam in May if they want to take this class. This course introduces limits and continuity, derivatives, the definite integral, differential equations, application of definite integrals, parametric, vector, and polar functions. After finishing this course, the student will enhance her critical thinking skills. Most of all they will definitely be ready for college calculus math. Students need to be familiar with AP central website to help them to prepare for the test.
Text: AP Calculus, Pearson-Prentice Hall (2007)
Accounting Philosophy - This course is designed to challenge students to meet their God given potential and to present opportunities for them to apply mathematical concepts to solve real-life problems.
Accounting I
This course is an introduction to the language of business. The students will be introduced to the planning, recording, analyzing, and interpreting of financial information so necessary in all aspects of business and in whatever environment they are in throughout their lives. Regardless of their responsibilities within an organization, the students will realize that they can perform their jobs more efficiently if they know the language of business.
Personal Finance Course Description: Personal Finance is based on use of an online course offered by the Universit y of Arizona. It uses newly developed research that can extend the abilit y to raise the personal financial capabilities of young adults. The students in this course will be able to create a personal financial plan, a personal budget and select strategies to use in handling credit cards and managing debt. Course Objective: Personal Finance prepares students for successful management of their
personal finances. It is a course that addresses the knowledge, skills, attitudes, and behaviors associated with the management of family economics and financial education.
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Just in Time Algebra for Students of Calculus in Management and the Lifesciences
Just-in-Time Algebra and Trigonometry : For Students of Calculus
Just-in-Time Algebra and Trigonometry for Calculus
Just-In-Time Algebra and Trigonometry for Early Transcendentals Calculus
Just-in-Time Algebra and Trigonometry for Early Transcendentals Calculus
Just-In-Time Algebra and Trigonometry for Students of Calculus
Mymathlab Mystatlab Student Access Kit For Ad Hoc Valuepacks
Summary
Strong algebra skills are crucial to success in applied calculus. This text is designed to bolster these skills while students study applied calculus. As students make their way through the calculus course, this supplemental text shows them the relevant algebra topics and points out potential problem spots. the table of contents is organized so that the algebra topics are arranged in the order in which they are needed for applied calculus.
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Within each curriculum organizer, students will be graded by learning outcome and assessed through tests, quizzes, assignments, and projects. Check my website for a list of the learning outcomes. If needed, students can retest certain learning outcomes provided they have completed all work for the unit and sought extra help. In the event of a retest, the second test will count for your grade. Retests must be arranged at a time of mutual convenience as soon as possible after the initial test. Failure to show up for a retest will result in the loss of the privilege. Students with unexcused absences will not be permitted to retest.
There will be a midterm exam worth 10% of the final grade, and a final exam worth 20% of the final grade.
Daily assignments will be assigned. Homework must be done regularly to be successful.
Topics Covered
o Rational Numbers
o Powers and Roots
o Equations and Inequalities (Algebra)
o Polynomials
o Linear and Non-Linear Relations (Graphing)
o Geometry
o Transformations
o Statistics and Probability
o Trigonometry (time permitting)
How to Succeed
1) Attendance: Be punctual! If you are late please knock once and wait to be let in.
2) Come prepared! Please bring a pen, pencil, paper, ruler and a calculator to every class.
3) Review your class notes. This will help you retain the information and help your performance on tests. It is also easier to stay up to date instead of trying to catch up.
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Contents
This course gives an introduction to the field of linear algebra. Concepts and techniques from linear algebra are of fundamental importance in many scientific disciplines and provide the "language" for understanding the behavior of
linear mappings and linear spaces. Topics covered are linear systems and Gauss method, vector spaces, linear maps and matrices, determinants, and eigenvectors and eigenvalues. In contrast to standard courses in linear algebra, we will combine the abstract concepts with application oriented examples in order to intensify the understanding in an algorithmically oriented way. Numerical methods for basic Linear Algebra problems will also be discussed.
Objectives
Learn linear algebra
Improve abstraction competences
Learn how to study on a math book
Learn about applications of Linear Algebra
Teaching mode
Lectures will be given following the standard systematisation of Linear Algebra. The lectures will mainly follow the book "Introduction to Linear Algebra" by Gilbert Strang. Moreover, the course will allow you to find and to use your favorite Linear Algebra book with profit - something we strongly advise you to do.
Tutorials will be given to discuss the weekly assignments.
References Introduction to Linear Algebra, by Gilbert Strang Introduction to Linear Algebra, by Serge Lang Video lectures by G. Strang at MIT A First Course in Linear Algebra a free book by Rob Beezer Linear Algebra a free book by Jim Hefferon
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More About
This Textbook
Overview
These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline
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The language is completely overcomplicated. The examples often leave out steps. The answer keys leave out even more steps.
Half of the stuff i remember from each lecture is worth entire chapters of the book. Which is, I suppose, why people go to college. Getting $130 per person per @#%$ book you put out, is why companies sell to america.
Cue highschoolers, 'lol thats why i never do homework or pass tests'
Cue people taking Statistics (I'm not), 'I so agree'Igon McAwg
10-27-03, 05:59 AM
What's really sad is I've had several math profs that were so bad that the book was the only way I learned anything.
Janen
EZ_Jasminne
10-27-03, 06:32 AM
i had one where 80% of the answers in the back of the book were wrong. it drove everyone crazy for a couple of weeks 'cause we'd never get the "right" answer until we finally realized what was going on.
EZ_Fallen One
10-27-03, 07:37 AM
i had a stats book last year that was edition...8...or somthing, and it had a bunch of new questions that edition 7 (or whatever was the prior edition) dind't. Unfortunately, about half of the new answers to the new questions never got changed, so the answers in the back of the book were half for edition 7. That was lots of fun.
As far as my calc went for me, i got most of my information out of class, i guess i'm more of a visual learner after seeing the steps repeated 1000 times. The 120$ book had a bunch of mojo that always either put me to sleep or scared me away from the text.
gogo college. "Pain inside is rising. I am the Fallen One. A figure in an old game. No jokers on my side. I've plunged into missery, i've turned out the light and murderd the dawn"
Krimzan
10-27-03, 07:43 AM
Engineering books are about the same. Paid like $140 for my Circuits book only to fail the class and say, "@#%$ this." God I hated that professor. "De boltage, de boltage is as a cONstant." That and saying 'study this for the test' and then testing us on something completly different is just low.
Llabak Tharr
10-27-03, 07:44 AM
I so agree
EZ_Swipey
10-27-03, 08:13 AM
EZ_Gyorg
10-27-03, 08:26 AM
What circuit book is that? I need a big, comprehensive circuits book just for home use. I had this one but it was the kind they only put the chapters they want in. Half the time I'd go to look for something the chapter wouldn't exist.
EZ_Talius
10-27-03, 09:19 AM
I'm taking Stats for psychology at the moment. It's quite a horrible experience. We don't have to know an equations (at least, not so far and it's been more than half the year) so basically all we're learning is the ideas behind stats (Programs do the rest of the work for us).
The textbook, considering it's almost 90% equations, examples of using the equations, and the practise questions to the equations, is completely useless.
Dragynphyre
10-27-03, 09:25 AM
Quote:I actually get angry when I realize that this is exactly what colleges do. The old 'look to you left, look to your right' speech, and all...
You can wind up thousands of dollars in debt for taking just one semester of 12 credits, and they're not even attempting to actually teach people who may need to pass that class for their degree?!?!
When you're paying upwards of $150 a credit hour, they should be teaching the class in such a manner that those who are willing/able to learn the material can understand at least 80% of the material.
No one should have to hire a tutor for even more money to teach someone what they should have been able to learn in the professor's class if the professor was not inept.Llabak Tharr
10-27-03, 10:23 AM
Quote:No one should have to hire a tutor for even more money to teach someone what they should have been able to learn in the professor's class if the professor was not inept.
Similarly, students should finally realize they're not going to learn anything in an hour 3 days a week. Class is the opportunity to discuss implications of what you've already learned and set the rough groundwork for what you're going to learn at home tonight.
Any college class that can 'teach' you everything you need to know just in the class time isn't worth taking.
EZ_Forceofmotion1
10-27-03, 10:24 AM
I have a big pet peeve with college math courses. I'm about as mathematically challenged as they get, but I still have to get through differential and integral calculus for my major. I look at the textbooks and weep in self-pity. (I'll be re-taking differential next semester in fact since I had to drop to audit this term. I've already purchased my copy of An Idiot's Guide to Calculus in the hopes it might improve my chances of passing.)
As for assistance from the lecture...all I can say is . If you're lucky enough to get a professor who speaks halfway intelligible english you've got 75% of the battle won there. (Of course you actually have to be lucky enough to have them assigned to your section, since most of the math courses at my university are listed as taught by STAFF.) Then you have to hope lightning strikes twice and they actually take the time to teach the material.
Dragynphyre
10-27-03, 11:08 AM
Quote:Any college class that can 'teach' you everything you need to know just in the class time isn't worth taking.
I understand that there is some work that the student themselves have to do - but at least the groundwork of the concept should be laid out in the class.
Most of the time in the classes I was attending, it was a struggle to even understand the language the professor was speaking in, let alone the concepts we were supposed to be introduced to...
Not to mention if they didn't move fast enough during the first hour of the lecture, they'd jam the other half of what they were going to go over into the last 15 minutes...
Hardly a good way to be introduced to a new concept, IMO.
Probably why I never got above a C in Calculus, and that was only because of partial work and the bell curve... I still can't do a derivative.
EZ_Prenn
10-27-03, 11:33 AM
Hah, how funny. Today we had a sub, she was asian (and frankly, enough people complain about the language barrier in science and math classes that it's a concern), but did a great job improvising with no notice.
And someone fell asleep, so we all got to laugh at him when she requested that somebody wake him up.
Nenjin
10-27-03, 11:36 AM
Quote:The examples often leave out steps. The answer keys leave out even more steps.
My biggest gripe with math books today. I NEED those steps shown to me in the equation!
Other than that, I do 90% of my learning in college out of my books. Lower level course lectures are for sux0rzyrkskog
10-27-03, 05:02 PM
Our Math Books are pretty international in Canada. There is someone from every single race in the book, and the word problems have bizarre foreign names no one has ever heard of. 'Hajaian dropped a basketball, after the 6th bounce, what is the total travel distance of the ball."
Madness I tell you!
EZ_Prenn
10-27-03, 05:55 PM
That sounds like it is ripped out of some comedy reutine, but it might just be because it's so true with modern text books.
Especially science books. The gay crippled korean girl with aids and the black doctor with heart murmors stand side by side for some obscure reason.
EZ_Kiltn
10-27-03, 06:23 PM
Quote:Especially science books. The gay crippled korean girl with aids and the black doctor with heart murmors stand side by side for some obscure reason.
Exactly
Inn I Evighetens Morke
EZ_Gyorg
10-27-03, 07:49 PM
The reasons they leave the steps out of the answer keys is so that professors can assign those problems. I've had multiple classes where we were given the answers. They graded us on our work inbetween.
Nenjin
10-27-03, 08:12 PM
That's fine, as long as they cover the inbetweens for us non-mathmatical rubes during class, which, they often do not.
EZ_Gyorg
10-27-03, 08:52 PM
From my experience thats in examples in the chapters.
EZ_Swipey
10-28-03, 06:01 AM
Sometimes it is, sometimes it isnt.
As I recall, the examples in the text are usually straight forward vanilla examples, and are directly applicable to maybe the first 4 problems in the 30 problem set at the end of the chapter. After those first 4, the problems start becoming other than vanilla number plugging exercises.
EZ_Gyorg
10-28-03, 06:23 AM
The idea of college is to create peiple able of solving problems. If you could just follow a cookie cutter prcoeedure for every problem you would be learning how to repeat steps, not how to solve problems. A professor I was at tango class with last weekend complained that her students never understand how to synthesize and come up with ways to solve problems slightly different from how they are explained in the book. The ability to just regurgitate facts and proceedures from the book just isn't enough.
Krimzan
10-28-03, 06:35 AM
Quote:
You can wind up thousands of dollars in debt for taking just one semester of 12 credits, and they're not even attempting to actually teach people who may need to pass that class for their degree?!?!
Yes. Depending on the major, in fact, there are classes which are...I hesitate to say designed, but designed to weed people out. I am convinced that there exists in the CS department at RIT a professor who is dedicated to wasting my time. I have come to the conclusion that I am not paying for an education, I am not paying to be taught, I am paying to get the piece of paper, that's it.
EZ_Swipey
10-28-03, 06:51 AM
Some people would argue that the idea of college is to create people who know things, first and foremost.
The history equivalent of a math book would be one that sketches out the causes and course of European colonialism in the new world to 1620, and then expects the students infer the American Revolution, Civil War, and eventual rise to superpower status. Those students who make the correct inferences (backed by accurate statements of cause and course) are encouraged to become history majors. Those who dont get bad grades and sneers from the more historically inclined.
The ability to ad-lib solutions to problems just isnt enough. One has to actually know things too.
Uh, in the levels of learning, regurgitating facts is like 3 levels lower than coming up with new, unique solutions. If you can come up with new answers, you can come up with the answers to just strait fact questions. Even history should be taught by the processes not the facts. Some things require knowing facts but in the real world, your always going to have access to the reference materials so you really need to be able to come up with answers that aren't in the book.
EZ_Swipey
10-28-03, 07:36 AM
History (at least at the (upper) college level) is taught as a process, not just as regurgitatable factoids. The thing is, the process needs the factoids to work, just as one needs to prime a pump before it will give water.
Due to the pathetic nature of our public schools, lower level history ourses are often what should be called remedial courses. They end up being 9 parts factoid to 1 part process. As the level of the course goes up, the factoid to process ratio changes until, eventually, the in class content becomes all process and no factoids (ie independent research)
People are complaining, I suppose, about the fact that college level math courses seem to have a much higher process to factoid ratio at the outset, which, due to the pathetic nature of our public schools mentioned earlier, about 90% of the students are unprepared for. Math departments, however, dont seem to give a flying *^&% about this, and are only interested in that 10% who, by nature or luck of the draw with their HS math teachers, are prepared.
PS : The numbers used (9-1 factoid to process, 90% unprepared, etc) are purely made up for emphasis, and in no way should one expect a citation to an unbiased, credible academic source for their origin.
Nenjin
10-28-03, 11:34 AM
Gyorg, you're right about what college is meant to do, put out people that can solve problems. And I guess it is my fault that I didn't keep a lot of my highschool mathmatics note, and the quadratic equation isn't permanently etched into my brain, or that I can't just remember the value of a function without working it out first. But colleges should understand this. Those first 4 vanilla problems are easily workable, but the rest of the chapter that requires critical thinking also requires that we as students have enough information to reference by. Asking people who aren't mathmatically inclined to make a leap in mathmatical logic is ab-@#%$-surd.
All I want is for those first 4 vanilla examples in the chapter to be systematically deconstructed, so I can see the process in it's entirety, go to the critical problems, and work backwards with all the information so til I'm comfortable enough with the concept I can think freely within it.
Teachers SHOULD be doing this as part of the cirriculum, but math teachers are(sorry to those that are) generally very intelligent, not very patient, and are not able to disseminate information to people who aren't good at math. I've had multiple teachers stare at me blankly because I don't get something, and they struggle for 5, 10, a semester to effectively get that idea across to me. It's a large part in me being dense to the material, but it is also about the teacher's being un-able to break it down, because they understand it so well, they breeze over things they think you, naturally, should get.
And before anyone wants to cluck about harder work, more attetion, stow it. If there is any subject in school I am riveted during class, and diligently do my homework, is math, and sometimes, it still doesn't make sense.
EZ_Gyorg
10-28-03, 11:44 AM
Have you ever gone in to see the professor during office hours?
EZ_Forceofmotion1
10-28-03, 11:58 AM
Right on, Nenjin.
That's exactly what mathematically adept people don't understand, and the reason I think that math departments have this problem. There are people who love the mathematical language and understand it with such innate fluidity that for them the "critical thinking" aspect is easy, because they know all the rules of the game and retain them, and they look down on people for whom it doesn't come so easily.
Just because someone has a hard time in math doesn't mean they have a problem in their ability to come up with solutions...I'm great at spatial puzzles and logic teasers...but I just don't retain or fully comprehend all the little foundational rules involved with math, and that's the kiss of death when facing a professor who doesn't take the time to break the steps down.
Trying to pass a college math class for people like me is like trying to win a chess game without being shown how to move your queen and knights.
Nenjin
10-28-03, 12:06 PM
Quote:Have you ever gone in to see the professor during office hours?
Can I count the ways for you?
Some teachers aren't bad, but even the nice ones have a hard time putting into the simplicities we need it to be in. And I'm not talking upper level stuff, I'm talking advanced algebra, logirithms, stuff like that.
EZ_Prenn
10-28-03, 12:39 PM
The most important thing about math is to understand the concept, not the procedure. When you realize what something is, it is easier to understand it. In my college algebra course, I see things that I had seen before, only now I am seeing what the application is and a bigger picture of what they are. Because the concept is typically NOT taught in math, it makes it hard for people who can't put two and two (figuratively) together. Once you understand the concept, either through blind happenstance or having it explained, it gets easier. Again, a failure of the institution.
The idea that it is to weed out the people who innately understand is flawed, because that is not teaching anything at all, even in the earliest classes. Which pretty much is the case, considering how many people have problems with advanced math concepts.
Expecting people to be creative and come up with their own solutions is fine. But, you can't expect someone to phonetically pronounce words without first knowing the alphabet.Evoll
11-02-03, 06:50 PM
Quote:Yes, there is one professor at my college like that. My current Math Proffessor is a 65+ year old grandmotherand probably the best math professor I have ever had. I had her for Calc 1/2 and I'm in her Calc 3 class now.
My old Anthropology Professor had a friend at another university. This friend was a high level mathmatics professor. The last few classes he was teaching before he retired consisted of this:
First day of class he would walk into the room and write an equation on the board. After doing that he would say "This is the only problem for this semester. If you finish it and you get an A, but more then likely none of you will be able to and will fail." Keric – ghay bard
Institute of PKER Reform
Decimators
EZ_Forceofmotion1
11-03-03, 01:11 PM
Hehehe, that's when you plug that puppy into Mathematica software.
EZ_Urusai
11-03-03, 02:47 PM
Yes you are just paying for the piece of paper.
I had taken Computer Science in collage as a major. They taught me nothing. Of all the classes I had the only ones that I really learned anything about computers in were the ones that didnt count towards my Major. (My Certified Novell Engineer Classes didnt count towards Computer Science Degree!?!?! ) That was just the big example, there were other smaller ones.
My third year we realized that the only different course between our Computer Science Degree and a Business Degree was ... get this ... "Basic Programming". Yes folks, a programming class (Which didnt even count as an elective towards the Computer Science Degree) was the only thing keeping the group of us from a Business degree as well. So what did some of us do, sign up for Basic Programming. It was obviously a joke class because half of us could write in basic before we fully understood the english language. The teacher was lost more often than we were. The simple "Ok today we are going to make "Hello" appear on our screens.
The teacher would lecture on for 30 minutes as to how we were supposed to accomplish this and our group would already be 'Tweaking' the thing. I personally made a "Hello" that rotated colors, bounced around the screen, while the computer played a clip of Bethoven that I happend to have the beep tone sequence for on disk. She scolded me for not writing that portion in class but gave me 100% for a grade anyway. The rest of the class was similar programming feats, the teach felt rather small because she was one of those that only knew it because she was teaching it.
I got off subject. Anyway, I went to admissions to get my degree. They told me that I couldent have it even though I qualified for all the course requirements because I didnt sign up for a business degree. They wanted me to pay 800 bucks or something. The sign-up fee for the degree then since I met all the requirements I would get the degree the NEXT semester. (I couldent even get it right then)
That was about the time my faith in the education system came to a crumbling pile of poo. I stopped going to collage and did some self study between work. With 3 months I had a job at a global company's IT Department. (I was lucky and found a manager that hired based upon skills and not papers. I blew away every other applicant, even the guy with 5 years of previous experience.)
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The materials below were developed for both graduate and undergraduate courses taught under the title "Mathematical Modeling." The philosphy of these courses is to cover as broad a range of topics with a modeling flavor as possible. It it generally easier to get a deeper knowledge of a subject X where one has seen the key ideas and some of the major results than it is to start off reading in a book devoted only to the content of topic X. Both issues involving modeling and situations where mathematics can be used to get insight will be considered. Some of these materials have been developed with the assistance of Stuart Weinberg, Teachers College.
Some of these items were also developed in conjunction with the P-credit courses offtered by Mathematics for America in conjunction with its Professional Development and Outreach in the New York City area.
Here are links to two very nice expository articles (pdf files) about how Gale/Shapley is used in the real world market of pairing medical students to hospitals where they can carry out there residency. The articles appeared original in SIAM News in 2003.
Here are two sets of "mechanical" exercises to cement your skill in solving bankruptcy problems and in using a variety of different methods to decide the winner of an election which involve preference ballots.
This is a brief list (with apologies to all the other wonderful books in this area) of important books about games, fairness, and elections. It also includes the important desirable features of a fair division scheme.
Mathematical modeling has interesting connections to other topics mathematicians and mathematics educators are interested in, namely, problem solving and estimation. In honor of July 4th here is a note with a poll and activity to probe the issues related to modeling, estimation, and problem solving.
There are many measures of central tendency in statistics (which single number "presents" a data set "best." Examples include the arithmetic mean, geometric mean, harmonic mean, mode, median, and mid-range value. When one wants to locate a facility (medical center, public lavoratory, fire house, etc.) one often desires to choose a "central" location so perhaps it may not be surprising that there are connections between statistics and facility location issues.
It may be worth the time to practice your understanding of the plurality, run-off, sequential run-off, Borda count, and Condorcet methods by doing the problems here. There are also questions that get at whether there might be "general theorems" involved in questions related to elections.
Elections and voting are the cornerstones of a democratic society. There are an amazingly large number of settings where American are involved with voting. We vote for local, state and federal officials, in some cases judges, and within our workplaces and educational institutions. There are also votes within legistlatures, clubs, and other "group structures." One can spend a whole lifetime studying mathematical insights into voting and elections. The mathematics involved covers the full gamut of mathematical tools.
In order to makes decisions one has to consider the various choices of actions that one can take and be able to understand which of these actions you consider as better. There are many aspects to this choice environment and this activity about probing your personal feelings about particular kinds of fruit my help see the complexities involved.
Here is a one one page introduction to the idea of a graph model. These are geometric diagrams which consist of dots (representing objects) and line segments which indicate relationships between the objects.
You can download a pdf file version of five modeling questions (each one of which can typically be solved in a single high school or college classroom period. The problems are based on the same "data" which consists of a grid graph with 6 sites singled out. The problems give rise (when scaled to realistic size, and to other settings) to heavily studied questions in "urban" operations research. You can find html versions of similar things if you scroll down to the urban operations research section below.
This activity is to suggest one of the many urban operations research problems which involve the routing of vehicles. This example centers around that gas must be delivered from a "depot" to the individual gasoline stations that get their gas supplies from this depot. This activity asks some questions but does not actually discuss the algorithms that have been developed to solve these kinds of problems. This activity is related to issues about bin packing and the traveling salesman problem.
This brief note discusses some ideas related Modeling Situation 1 and the modeling process in general. Specifically, the issue of finding data and information that is used in model construction is raised.
This brief note discusses some ideas related Modeling Situation 2. Different election decision methods give rise to different winners and this means that one has to try to think of ways to assess the pros and cons of different methods. Many nice lessons for K-12 education can be based on election methods and voting ideas. When there is no Condorcet winner for an election what should be done to make the election method "decisive." When there is a Condorcet winner is this the best method? There is also the chance to practice the construction of the pairwise preference matrix and the anti-plurality method is mentioned.
In many social choice and game theory settings there are payoffs for the outcomes to the participants. While these payoffs sometimes can be thought of as money, psychologically the same amount of money can mean different things to different people. The concept of tility is an attempt to deal with these complexities.
This note talks briefly about Arrow's Theorem, fairness axioms for election decision methods, and strategic voting. Strategic voting refers to voting for a ballot which reflects something other than your sincere feelings because it will give you a more favored outcome. This can be done when you know what the decision method being used is, and when data about how other voters might vote is available.
This note talks about an additional fairness condition, called Majority, that some would say should be obeyed by a "reasonable" election decision method. Although much more could be said about "election decision methods" we will move on to another phenomenon that occurs in voting situations. Namely, that votes are being taken by "players" (legislators) who represent groups of different population or economic power. To deal with situations of this kind, weighted voting is sometimes used. The idea is to have each player cast a "block" of votes at once, called the weight of the player. This situation arises in the Electoral College and many of the governing bodies of the European Union.
Here you can get a chance to practice problems involving the concepts of winning and minimal winning coalitons in a weighted voting game, as well as computing the Banzhaf and Shapley voting power for a weighted voting game.
This note has all the inequivalent wieghted voting games (which make some sense in practice) with 4 or fewer players. The weighted voting, minimal winning coalitons, and Shapely-Shubik power vector for each game is given.
This note proposes an "open question" which may be of interest to you, or if you teach, to your students. The question involves representing weighted voting games in a way that shows the Banzhaf power relations to the players.
Some ideas about weighted voting are given here, including the definitions and statement of the theorem of Alan Taylor and William Zwicker about when a voting game can be represented by a weighted voting game. The idea involves the trading of players between winning coalitions. Also, basic ideas about the apportionment model are developed. How should an integer number of seats, which must be assigned to claimants in integer amounts, be assigned based on the size of the claims put forward by the claimants?
This note has some information about apportionment models in classical (how many seats does a party get in a parliament based on the votes for the parties) as well as other settings. It is important that apportionment methods be viewed as fair which requires a way to judge whether or not one apportionment is better than another.
Three problems are posed involving payoffs in a two player game, where each player can choose two actings, and where the payoffs to the players add to zero. Our goal is to try to determine when a game of this kind is fair.
This note offers extensive references on apportionment problems as well as specific examples showing how different measures of optimality can be used to defend different apportionment methods. The approach developed here is whether or not the transfer of a seat from one state to another makes the measure smaller or bigger. A very different approach is a global optimization approach.
These are notes about games in extensive form which offer lots of ways to model conflict situations, including games with a dynamical quality (e.g. reactions of what player 2 can do to what player 1 has done). A brief discussion of Rheinhard Selten's work on extending that of John Nash is given.
Some notes about the problem of distributing a quantity E to claimants whose claims exceed E. Bankruptcy like problems arise in the distribution of water, or emergency funds, as well as problems concerning the funding of an amount E by collecting taxes from different income groups.
These situations give rise to many important problems in graph theory, operations research, and other parts of mathematics. For notes about what the key ideas are that these problems lead to, look at the following:
For the situations 5 and 6 which are not mentioned in the document above, the relevant notes are: Situation 5, Voronoi diagrams; Situation 6, Robbin's Theorem concerning when it is possible to orient a connected underdirected graph, so that the resulting digraph becomes strongly connected.
This election example introduces a notation due to Duncan Black (British political scientist) for expressing preferences of a "voter" in a situation where choices must be ranked by an individual voter. Choices listed towards the top are preferred by the voter.
This glossary offers a variety of terms that arise in the use of mathematics to study fairness questions. The terms are drawn from social choice theory (voting and elections), apportionment, and other domains.
One of the most remarkable theorems that mathematics has contributed as an insight into political science and economics is a result of Kenneth Arrow, who won the Nobel Prize for his work. The basic result is that for decision methods that produce rankings when there are three or more alternatives, there is no decision method which obeys a short list of reasonable "fairness" conditions.
An apportionment example using Webster, Jefferson's and Adam's Methods is worked out, Using the "divisor" approach to these methods. For relatively small house sizes one can usually do problems of this kind using "rank index" approaches to divisor methods.
Here is a sampler of fairness and equity problems for a general audience or to introduce a class in middle school or high school to some problems that lie within the domain of what mathematicians are studying that are related to fairness. Fairness Sampler
This essay discusses some mathematical models in political science. The public is accustomed to the effectiveness of using mathematics in physics, engineering, chemistry and biology. Howver, the fact that mathematics provides major insights into the social sciences is less appreciated.
g. Connection between facility location and statistics (This essay which supports the previous notes, deals with ideas connecting facility location problems and statistical concepts such as the mean, median, and mode. The audience is student in grades 6 and higher.)
Fairness Models
a. Voting and Elections (This is a primer about the history of mathematical insights into elections and voting systems.)
b. Apportionment I (This is the first of a two part essay about apportionment problems, such as deciding how many seats to give each US state in the House of Representatives based on the populations of the states.)
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You can find a broad list of topics dealing with fairness in this "syllabus" for a college level "humanities course" about fairness.
In the United States the President is not elected directly through popular vote but using the Electoral College. The Electoral College has 51 players who cast different numbers of votes, loosely proportional to the population of the regions the "electors" represent. Here is an example to show that a naive way of assigning weights to voters in a weighted voting game can result in "players" who have positive weight but no power!
General Modeling
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The Consortium for Mathematics and Its Applications has produced a wide array of excellent materials that deal with all aspects of Mathematical Modeling.
This includes many modules, and a journal (UMAP Journal) devoted to both research about mathematical modeling and educational issues related to mathematical modeling. Its "membership privilege" journal, published twice a year has lots of articles and materials about modeling.
A wide variety of mathematical modeling problems at various levels of difficulty and with a variety of levels are available on the web. There are also a variety of contests for high school students and college students about mathematical modeling. Some of these contests are run by COMAP and another by SIAM, the Society for Industrial and Applied Mathematics.
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Computer aided mathematics teaching
People
Brief description
The primary goal of this research group is to develop materials, technologies and methods to improve mathematics teaching and learning outcomes in the Aalto University School of Science. In order to support this goal, we research e-learning methodologies in mathematics. We are also actively involved in international collaboration and open source software development. For further information about our project and activities, see our portal, intmath.org.
Projects
We are taking part in Support Successful Student Mobility with MUMIE project, which is funded with support from the Lifelong Learning Programme of the European Union. More about the project:
Linda Havola: Improving the teaching of engineering mathematics: a research plan and work in-process report. In proceedings of the Joint International IGIP-SEFI Annual Conference 2010. Trnava, Slovakia, 2010.
pdf
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It's a pilot project, so it remains to be seen if this approach makes a difference in improving the pass rates for students in lower-level math courses like College Algebra, which have been at around 60…
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Summary
4-color hardback text w/complete text-specific instructor and student print/media supplement package AMATYC/NCTM Standards of Content and Pedagogy integrated in Exercise Sets, Sourced-Data Applications (students are also asked to generate and interpret data), Scientific and Graphing Calculator Explorations Boxes, Mental Math exercises, Conceptual and Writing exercises, geometric concepts, Group Activities, Chapter Highlights, Chapter Reviews, Chapter Tests, and Cumulative Reviews 6 step Problem-Solving Approach introduced in Chapter 2 and reinforced throughout the text in applications and exercises helps students tackle a wide range of problems Early and intuitive introduction to the concept of graphing reinforced with bar charts, line graphs, calculator screens, application illustrations and exercise sets. Emphasis on the notion of paired data in Chapters 1 and 2 leads naturally to the concepts of ordered pair and the rectangular coordinate system introduced in Chapter 3. Graphing and concepts of graphing linear equations such as slope and intercepts reinforced through exercise sets in subsequent chapters, preparing students for equations of lines in Chapter 7
Table of Contents
Preface
ix
Review of Real Numbers
1
(72)
Symbols and Sets of Numbers
2
(9)
Fractions
11
(8)
Exponents and Order of Operations
19
(6)
Introduction to Variable Expressions and Equations
25
(5)
Adding Real Numbers
30
(6)
Subtracting Real Numbers
36
(5)
Multiplying and Dividing Real Numbers
41
(7)
Properties of Real Numbers
48
(7)
Reading Graphs
55
(18)
Group Activity: Creating and Interpreting Graphs
62
(1)
Highlights
63
(4)
Review
67
(3)
Test
70
(3)
Equations, Inequalities, And Problem Solving
73
(90)
Simplifying Algebraic Expressions
74
(7)
The Addition Property of Equality
81
(7)
The Multiplication Property of Equality
88
(6)
Solving Linear Equations
94
(10)
An Introduction to Problem Solving
104
(8)
Formulas and Problem Solving
112
(10)
Percent and Problem Solving
122
(9)
Further Problem Solving
131
(8)
Solving Linear Inequalities
139
(24)
Group Activity: Calculating Price Per Unit
149
(1)
Highlights
150
(6)
Review
156
(3)
Test
159
(1)
Cumulative Review
160
(3)
Graphing
163
(64)
The Rectangular Coordinate System
164
(11)
Graphing Linear Equations
175
(10)
Intercepts
185
(10)
Slope
195
(13)
Graphing Linear Inequalities
208
(19)
Group Activity: Financial Analysis
217
(1)
Highlights
218
(4)
Review
222
(2)
Test
224
(1)
Cumulative Review
225
(2)
Exponents And Polynomials
227
(54)
Exponents
228
(10)
Adding and Subtracting Polynomials
238
(8)
Multiplying Polynomials
246
(6)
Special Products
252
(5)
Negative Exponents and Scientific Notation
257
(9)
Division of Polynomials
266
(15)
Group Activity: Making Predictions Based on Historical Data
273
(1)
Highlights
274
(2)
Review
276
(2)
Test
278
(1)
Cumulative Review
279
(2)
Factoring Polynomials
281
(62)
The Greatest Common Factor and Factoring by Grouping
282
(7)
Factoring Trinomials of the Form x2 + bx + c
289
(6)
Factoring Trinomials of the Form ax2 + bx + c
295
(9)
Factoring Binomials
304
(5)
Choosing a Factoring Strategy
309
(5)
Solving Quadratic Equations by Factoring
314
(10)
Quadratic Equations and Problem Solving
324
(19)
Group Activity: Choosing Among Building Options
333
(1)
Highlights
334
(3)
Review
337
(2)
Test
339
(1)
Cumulative Review
340
(3)
Rational Expressions
343
(68)
Simplifying Rational Expressions
344
(7)
Multiplying and Dividing Rational Expressions
351
(5)
Adding and Subtracting Rational Expressions with Common Denominators and Least Common Denominator
356
(7)
Adding and Subtracting Rational Expressions with Unlike Denominators
363
(6)
Simplifying Complex Fractions
369
(7)
Solving Equations Containing Rational Expressions
376
(7)
Ratio and Proportion
383
(7)
Rational Equations and Problem Solving
390
(21)
Group Activity: Comparing Formulas for Doses of Medication
399
(1)
Highlights
400
(5)
Review
405
(2)
Test
407
(1)
Cumulative Review
408
(3)
Further Graphing
411
(42)
The Slope-Intercept Form
412
(6)
The Point-Slope Form
418
(7)
Graphing Nonlinear Equations
425
(8)
Functions
433
(20)
Group Activity: Matching Descriptions of Linear Data to Their Equations and Graphs
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Math Word Problems Demystified 2/E
Find yourself stuck on the tracks when two trains are traveling at different speeds? Help has arrived! Math Word Problems Demystified, Second Edition is your ticket to problem-solving success.
Based on mathematician George Polya's proven four-step process, this practical guide helps you master the basic procedures and develop a plan of action you can use to solve many different types of word problems. Tips for using systems of equations and quadratic equations are included. Detailed examples and concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.
It's a no-brainer! You'll learn to solve:
Decimal, fraction, and percent problems
Proportion and formula problems
Number and digit problems
Distance and mixture problems
Finance, lever, and work problems
Geometry, probability, and statistics problems
Simple enough for a beginner, but challenging enough for an advanced student, Math Word Problems Demystified, Second Edition helps you master this essential mathematics skill.
The premise behind Daily Word Problems is simple and straightforward-frequent, focused practice leads to mastery and retention of the skills practiced. When you guide your students in solving a word ...
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Learning Goals & Objectives
Student Learning Goal 1:
Mathematics majors will perform tasks requiring logical reasoning.
Students will:
Objective A: Understand the different types of mathematical statements and how they are used, including definitions, axioms, hypotheses, conclusions, theorems, corollaries, lemmata, and conjectures;
Objective B: Understand methods of proof.
Student Learning Goal 2
Mathematics majors will know the content of the fundamental fields of mathematics and can perform tasks requiring complex reasoning.
Students will:
Objective A: Use basic skills to manipulate expressions;
Objective B: Know the basic definitions and theorems of mathematics;
Objective C: Be able to perform tasks requiring complex reasoning.
Student Learning Goal 3
Mathematics majors will advance their understanding and knowledge of mathematics and their ability to convey mathematical concepts through currently available technology.
Students will:
Objective A: Information Literacy: Students will use the internet and/or library resources to obtain relevant information concerning historical information or mathematical content in regards to current course or project. Students will be expected to both look up sources and learn to search for their own sources.
Objective B: Computation: Students will use computers or graphing calculators to perform labor-intensive calculations and/or create graphical displays. Programs include, but are not limited to, Excel, Minitab (or other statistical software), and Mathematica (or other software).
Objective C: Presentation: Students will use technology for the purpose of elegantly presenting mathematical ideas, theories or results. Technologies include PowerPoint, Prezzi, Jing, Beamer, Latex, Word, and graphical tools.
Student Learning Goal 4
Mathematics majors will communicate mathematical ideas with precision and clarity.
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This objective focuses on showing understanding of situations by describing them mathematically and by making correct use of symbols, words, diagrams, tables and graphs Using and applying…
This objective focuses on beginning to organise work, and using and interpreting mathematical progression…
This objective focuses on representing problems and synthesising information in algebraic, geometric or graphical form; and moving from one form of presentation to another to gain a different perspective on the problem/task…
This objective focuses on discussing work using mathematical language and representing work using symbols and simple…
This objective focuses on interpreting, discussing and synthesising information presented in a variety of mathematical forms, and beginning to explain reasons for selection and use of
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The Quadratic Calculator together with a comprehensive package of instructional study aids is an invaluable educational resource for students and teachers alike in mathematics, science, engineering and finance.
The Quadratic Calculator is easy to use and provides solutions to quadratic functions and related quadratic equations, as well as the solutions needed to graph the quadratic function.
These solutions include the real and imaginary roots, the discriminant, the maximum or minimum value of the quadratic function, the sum and product of the roots and the related vertex form of the quadratic function.
The complementary study aids are presented in a compact e-reference book that includes sections on fundamental concepts, formulas, problem-solving methods, tips for sketching a quadratic function, together with numerous worked step-by-step example and applied problem sets. There is even a summary quiz with answers that tests your knowledge and understanding of the quadratic function and equation.
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2.Overview of How Maple enhances the
student'slearning in Multivariable
Calculus
3.Illustration and practice with major features
for Multivariable Calculus through example Maple worksheets for the classroom
4.Student Multivariable Calculus Projects in
Maple
Introduction
I
have been teaching our second year (Calculus III and IV) for the last 13 of my
27 years at Saint Joseph's
College.I first became acquainted with
Maple at one of the very early ICTCM conferences.We first adopted Maple V to use in Calculus
and other higher level courses.We are
using Maple 12 this year and will likely upgrade versions next year.The Calculus III and IV classes meet three
times a week.One of those three
meetings is in a computer lab.During
the other two class times, we are in a regular classroom equipped with a
computer with Maple, and projection.
The
four primary areas where we have seen advantages in using Maple for our second
year calculus are:
Computation:For some concepts, such as tangent and normal
vectors, computations can be very cumbersome and thus interfere with conceptual
understanding.Maple can remove the drudgery
of the computations, allowing students to focus on theory, methods, and
applications.
Assignment
Verification (Checking Answers):We still want out students to be able to
carry out computations(show
work!).However,Maple can perform the step by step
calculations as well, giving the students a chance to check their work and find
their errors.
Independent
Exploration and Student Projects:Maple's interface allows students to explore and
write about concepts within the same document.They can easily edittheir Maple
commands as well as their accompanying writing.Maple 2D math encourages student to write their mathematics with correct
mathematical notation.For projects,
the interactive style of the software allows students to start by implementing
a scaled-back portion of a project idea, then iteratively expand upon that
idea.By using Maple's help facilities
with instructor's advice and trouble-shooting, students can develop projects
that are really fun as well as educational.
At our annual Saint
Joseph's College Undergraduate Colloquium, Calculus
IV students have been presenting their projects in an informal walk-through
"poster session".This format gives them
a chance to explain their work to their peers and appreciative faculty in very
creative ways.Traditionally they have
shown their Maple worksheet interactively along with creative posters or a
slide show presentation.They often
have a fun activity for the attendees.Some of the memorable activities include:
Maple
is a real power tool, which means it requires an investment of time to learn to
use its features effectively.Then -- as
soon as one starts to feel comfortable – an upgrade is released with new
features to master.Some of the helpful
resources for keeping up are:
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Event Description
Composition is hard for students -- but the multiple representations shown in this webinar will get them over the rough spots and help them to generalize and master the concept of composition. We'll compose functions both geometrically and symbolically, find the links between different representations, and put the behavior of the variables front and center. We'll show, and make available to attendees, composition activities that are accessible to early-algebra middle-school students and activities that will give high school algebra and pre-calc students new insights. The function dance activities from this webinar (in which the dependent variable dances with the independent variable) are likely to become a favorite for students at all levels.
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High school math for grade 10, 11 and 12 math questions and problems to test deep understanding of math concepts and computational procedures are presented. Answers to the questions are provided and located at the end of each page.
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Describes the hidden human, emotional, and social forces that shape mathematics and affect the experiences of students and mathematicians. This book is suitable for those who want to understand why the most rational of human endeavors is at the same time one of the most emotional.
The Pythagorean Theorem is one of the best-known equations in mathematics. Its origins reach back to the beginnings of civilization. What most non-mathematicians don't understand or appreciate is why this simply stated theorem has fascinated countless generations. This book explores the history and importance of this remarkable equation.
Get the confidence and math skills you need to get started with calculus Are you preparing for calculus? This hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in the course.
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Practical Problems in Mathematics for Welders, 6th Edition
ISBN10: 1-111-31359-8
ISBN13: 978-1-111-31359-3
AUTHORS: Chasan
Discover how this highly effective, practical approach to mathematics can prepare you with the math skills most important for success in today's welding careers. PRACTICAL PROBLEMS IN MATHEMATICS FOR WELDERS, 6E combines an inviting, comprehensive introduction to math with an emphasis on the latest procedures, practices, and technologies in today's welding industry. You'll see how welders rely on mathematical skills to solve both everyday and more challenging problems, from measuring materials for cutting and assembling to effectively and economically ordering materials. Highly readable, inviting units emphasize both basic procedures and more advanced mathematics formulas welders must regularly use.
Clear, uncomplicated explanations, new practice problems, and real-world examples emphasize some of the industry's latest developments. New, more dimensional illustrations throughout this edition help you further visualize concepts with an approach that's ideal for learners at all levels of math proficiency. A new homework solution and dynamic online website help reinforce the math skills most important for success in welding
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Differential Equations
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODEs) deal with functions of one variable, which can often be thought of as time. Topics include: Solution of first-order ODE's by analytical, graphical and numerical methods; Linear ODE's, especially second order with constant coefficients; Undetermined coefficients and variation of parameters; Sinusoidal and exponential signals: oscillations, damping, resonance; Complex numbers and exponentials; Fourier series, periodic solutions; Delta functions, convolution, and Laplace transform methods; Matrix and first order linear systems: eigenvalues and eigenvectors; and Non-linear autonomous systems: critical point analysis and phase plane diagrams
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2249937 / ISBN-13: 9780582249936
Introduction to graph theory
In recent years graph theory has emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as ...Show synopsis mathematics, computer science and economics, and as a readable introduction to the subject for non-mathematicians. The opening chapters provide a basic foundation course, containing definitions and examples, connectedness, Eulerian and Hamiltonian paths and cycles, and trees, with a range of applications. This is followed by two chapters on planar graphs and colouring, with special reference to the four-colour theorem. The next chapter deals with transversal theory and connectivity, with applications to network flows. A final chapter on matroid theory ties together material from earlier chapters, and an appendix discusses algorithms and their efficiency.Hide synopsis
Graph Theory has recently emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology and genetics. This book provides a comprehensive introduction to the subject.Graph Theory has recently emerged as a subject in its own right, as well as being an important mathematical tool in such diverse subjects as operational research, chemistry, sociology and genetics. This book provides a comprehensive introduction to the subject582249937 Brand New Paperback Overseas International...New. 0582249937
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This text is a practical course in complex calculus that covers the applications, but does not assume the full rigour of a real analysis background. Topics covered include algebraic and geometric aspects of complex numbers, differentiation, contour more...
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Math Behind the Science summer bridge program
Mission:
The Mathematics Behind the Science summer bridge program is designed to enhance the mathematics preparedness of college-bound students who are interested in pursuing careers in science, mathematics, engineering, technology, or medicine. Specifically, it will prepare students to enter the introductory calculus course and provide a foundation for success in that important course.
Program Description:
Mathematics Behind the Scienceis a STEM (Science, Technology, Engineering, Mathematics) college preparation program that serves students who have completed grade 12 and will enter university as freshmen in the fall, or students already attending UVI who have successfully completed MAT 024. It is intended to serve as a bridge between high school curriculum and expectations at university.
The courses are designed to show students some of the ways that mathematics and computation are used in scientific contexts and to enhance the preparedness of college-bound students who are interested in pursuing STEM careers. The courses will provide a foundation for success in STEM courses in the freshman year. Mathematics is required for majors in both Mathematics and Applied Mathematics (Dual Degree Engineering Program); it is also required for Bachelors degrees in Biology, Marine Biology, Chemistry, Computer Science, and the Associates degree in Physics.
Mathematics Behind the Science a residential program. St. Thomas, St. John and St. Croix students may reside in the dorms on the St. Thomas campus. The program will meet every day, Monday through Friday for six weeks in the summer (June-July).
For additional details, a Mathematics Behind the Science Summer Bridge Program Flyer is available under our Brochurespage. Feel free to download and share this document.
Purposes of Mathematics Behind the Science summer bridge program:
·To illuminate uses of mathematics and computation in the sciences and strengthen students' ability to use computation as a problem solving tool in future careers.
·To strengthen the mathematical proficiency of entering students, better preparing them for success in college level mathematics courses.
·To facilitate the transition between high school performance and that expected at the university level.
·To establish a community of learners supporting student success throughout the undergraduate years as science STEM major.
What are the components?
·A unique mathematics program exploring the mathematical concepts that underpin the concepts of calculus. Significant connections will be made to biology, chemistry, physics and the environment.
·A course in computing that will focus on introductory computer science skills, with applications to the sciences in order to prepare students for a college STEM curriculum and today's environment where computation is used in Biology, Chemistry, Engineering, and other related STEM disciplines.
·A Success in STEM course to give students information that will help them to transition to college life, including seminars related to use of library, information technology, student services, and scholarship programs to support student success at UVI.
·A Scientific Reading and Writing course.
·Career sessions conducted by practicing scientists and field trips to demonstrate mathematical modeling in the Caribbean environment.
Eligibility:
The applicant must have:
·Genuine interest in pursuing a program of study in science, mathematics, engineering, technology, pre-medical or other health professional field;
·acceptance into an accredited college or university forstudents in grade 12(preference given to applicants enrolling at UVI)
There is no cost to participants or their parents. Tuition will be provided free by the University. Students will be provided with lunch in the UVI Cafeteria. No stipend will be provided.
How does one apply for admission?
The Mathematics Behind the ScienceApplication is available for download as a PDF or Word document from our Forms page (To learn more, we encourage you to download the program flyer from our Brochurespage.) A hard copy of the application form may be obtained by contacting Ms. Aimee Sanchez at 340-693-1249 (Email: aimee.sanchez@uvi.edu, Office Location: UVI Classroom Administration Building Room CA 202).
Program Director:
Robert Stolz, PhD College of Science and Mathematics University of the Virgin Islands St. Thomas, VI 00802 Email: rstolz@uvi.edu Office: 340-693-1231 Fax: 340-693-1245
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Discrete Mathematics
9780198534273
ISBN:
0198534272
Publisher: Oxford University Press, Incorporated
Summary: This text is a carefully structured, coherent, and comprehensive course of discrete mathematics. The approach is traditional, deductive, and straightforward, with no unnecessary abstraction. It is self-contained including all the fundamental ideas in the field. It can be approached by anyone with basic competence in arithmetic and experience of simple algebraic manipulations. Students of computer science whose curric...ulum may not allow the study of many ancillary mathematics courses will find it particularly useful. Mathematics students seeking a first approach to courses such as graph theory, combinatorics, number theory, coding theory, combinatorial optimization, and abstract algebra will also enjoy a clear introduction to these more specialized fields. The main changes to this new edition are to present descriptions of numerous algorithms on a form close to that of a real programming language. The aim is to enable students to develop practical programs from the design of algorithms. Students of mathematics and computer science seeking an eloquent introduction to discrete mathematics will be pleased by this work
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Calculus Text Puzzles
This is the home page for Calculus Text Puzzles: An interactive way of
reading scrambled calculus definitions, examples and proofs. Some
parts of the text on these dynamic webpages can be rearranged easily,
and the reader is given automatic feedback on whether the original
correct form of the text has been
reconstructed.
Click on some of the links below and see how this approach
allows you to (re)discover concepts. This can help with understanding
calculus, and it is more interesting than simply reading the textbook.
It is meant as an intermediate step between studying existing
definitions and proofs versus writing your own. You are encouraged to print out
the unscrambled version of a completed puzzle and keep it as a
study-aid. The pages below require a fairly recent web
browser such as Internet Explorer 5.5+ or Netscape 6+. If you have an
older browser, you may be able to view the previous version of
Calculus Text Puzzles.
Instructions: Some pieces of the definitions and results below
have been shuffled, and your task is to sort them into the logically
correct order (if there seem to be several correct orders, choose one
that produces the 'most sensible' result).
Click a puzzle piece
to select it,
then click another puzzle piece
to move (or swap) the selected piece to
that particular position. When you think you are done,
click on the Check button.
If you are happy with your grade, you can enter your name and
print the page. Don't worry, the grade and your name are not recorded
anywhere on the web (the Text Puzzles run locally in your browser).
Chapter 4: Applications of Derivatives
Definitions:
Concepts of derivatives related to finding maxima and minima.
(for Netscape 7 or IE+MathPlayer)
Results:
The Mean Value Theorem and some of its consequences.
(for Netscape 7 or IE+MathPlayer)
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...this text has the advantage of being self contained: the conceptual prerequisites are kept to a minimum and the student can quickly get to the applications with a reasonably complete understanding of how and why wavelets work in data processing. As such, this text should appeal to instructors and students who are interested in learning about how wavelets work without being required to learn the whole mathematical backdrop in which wavelet theory is developed. (Zentralblatt Math, Volume 940, No 15, 2000)
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Workshops Here you can find complete tutorials, for example how to build a tree, creation of flowers and more.
Functions
Boys and girls, please open your math books! Many people fear the use of equations and functions. This chapter will try to show you how to make use of equations – even if you don´t like math and science.
FAQ – most frequently asked questions Just like the title says: this chapter will try to cover the most frequently asked questions.
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Mathematics
Need to figure out how many overtime hours you must work to make up the lost wages from an unexpected sick day? Want to alter grandma's lasagna recipe to serve 10 instead of 8? Is Pi Day your favorite holiday? If so, then you might want to figure math into your education equation. Mathematics is the science of numbers. Mathematicians use patterns and symbols to formulate and test theories.
As a BYU undergrad, you will have access to top-notch facilities and equipment. Not only is our gear great, but our staff and faculty are as well. In the Math Lab, you can ask skilled student employees for help in any of your mathematics courses. Professors and TAs will be available every step of the way to help guide you on your quest to become a mathMathematics is the search for truth and understanding. It is painting a picture of reality with symbols."
Riding Mathematical Waves
Options for Retailers
Undergraduate research makes classroom learning come alive and can help propel you into a professional career. You will have the opportunity to put your book knowledge into practice by working on real research projects. The Math Department has several special labs and resources available to students including Interdisciplinary Mentoring Program in Analysis, Computation and Theory (IMPACT), Summer Research Experience for Undergraduates (REU), and Center for Mentoring Undergraduate Research in Mathematics (CURM). We teach students technical and research skills that will prepare them areas include making and breaking codes, creating techniques to model sound waves, and using math to explore the many dimensions of the universe.
• Algebraic Geometry
Students study curves, surfaces, and other shapes defined by systems of algebraic equations. This combination of algebra and geometry has many areas of application including computer graphics, cryptography, and mathematical physics.
• Applied Mathematics & Mathematical Physics
Applied mathematics students research mathematical methods that are used in science, engineering, business, and industry. It describes the professional specialty in which mathematicians solve practical problems.
Mathematical physics students study the development of mathematical methods that are applied to physics. It develops methods suitable for the formulation of physical theories.
• Combinatorics & Matrix Theory
Combinatorics students study finite structures. Aspects include counting the structures, deciding when criteria can be met, and constructing and analyzing objects meeting certain criteria.
Matrix theory students study matrices; rectangular arrays of numbers, symbols, or expressions. Matrices are used in most scientific fields including physics, computer graphics, and quantum mechanics.
• Differential Equations & Dynamical Systems
Students research systems that evolve in time, with a particular focus on how short-term rates of change affect long-term outcomes. This theory is applied to the study of many things including the motion of the solar system, the growth of populations, and the spread of disease.
• Geometric Topology & Geometric Group Theory
Geometric topology students research settings which mathematicians call a space. This branch of math emphasizes aspects that are most closely allied to classical geometry like distances, polyhedral objects that generalize intervals and triangles; manifolds that generalize planes, surfaces, and Euclidean spaces.
Geometric group theory students research models of symmetries and rigid motions. It is applied when objects such as a molecule or a space exhibits symmetries or admits motions. The group of symmetries or motions reflects the geometry of the object or space on which it acts.
• Mathematical Biology
Students focus on the mathematical representation, treatment, and modeling of biological processes. This branch of math has a variety of applications ranging from predicting how populations change over time, how infectious diseases spread, and how do cells move.
Students study equations involving unknown functions and several independent variables and their partial derivatives. These equations are used to solve problems such as the propagation of sound, heat, or elasticity.
• Stochastic Differential Equations
Students research equations in which one or more of the terms is a stochastic process. SDE are used to model various phenomena from fluctuating stock prices to thermal fluctuations.
Many of our students go on to get advanced degrees and additional experience that broaden their career opportunities. BYU Mathematics alumni have found jobs in government, academia, and numerous business positions including working at:
"Math is used all over the place—in finances, insurance, engineering, physics, and several other industries. Taking math classes will help you think more analytically and help you to be more diverse in life."
–Darrell Johnson, Symetra Financial
Eric Murphy Lecture
Hear the confessions of a recovering English major and unrepentant math nerd.
Hands On: Riding Mathematical Waves
Join the Hands On team and the Department of Mathematics as they figure out how waves work.
Options for Retailers
See how a BYU student developed options for retailers to help protect against risks.
We Use Math
Discover the opportunities and success you can have by studying math.
Why Mathematicians Play with Bubbles
See math students determine what shape of bubble has the least surface area.
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give further examples and methods to strengthen and extend your algebraic
way of writing and reasoning. Rules are introduced as they are needed. The fifth chapter
Arithmetic Rules and Patterns for Algebra (algebraically described)
describes how arithmetic properties, more precisely assumptions about
arithmetic, can be employed in algebra along with the ideas of
replacement and substitution. The properties are described symbolically.
The previous chapters should give readers a mastery of the algebraic way
of writing and thinking sufficient to understand the symbolic or
algebraic statement of the properties or rules.
The chapters can be read out of order if you wish. Some chapters, not
always the first, may be easier than others. Remember what is not
immediately understood by yourself might be well-known or easily
understood by another.
A Word of Caution
The following chapters show how algebraic way of writing and reasoning
solves many problems of a similar type. Their solutions follow the same
arithmetic or algebraic pattern. Yet a caution is required here. Once the
derivation of a formula or the algebraic pattern of a solution was
understood, I once thought computational examples to be a waste of time
and energy — a distraction from the mastery of further algebraic
patterns. To some extent this is true. But in physics, chemistry and
business, a familiarity with the size and magnitudes of the numbers and
quantities appearing in calculations is needed to avoid or spot simple
numerical errors or hazards. This familiarity is only provided by doing
some problems. Doing too many problems may leave insufficient time to
learn and master the algebraic reasoning in a subject
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tion reform. A parent armed with the advice in this book could do a lot to help improve a child's education.?Amy Brunvand, Univ. of Utah Lib., Salt Lake CityCopyright 1998 Reed Business Information, Inc.
Reviewed by aardwolf11, (Calgary, AB)
athematical journey".
Reviewed by "gloriungus", (albuquerque, nm)
kind of results we want from their students.
Reviewed by Daryl R. Anderson "dander", (Trumansburg, NY USA)
You've probably heard that youngsters who are anxious about math also do poorly in math. A lot of folks thought this was just because students with limited ability appropriately worried about the subject. Not so!Just the other day I clipp
Reviewed by "mangelone", (NUTLEY, NJ USA)
When I think of Dr. Patricia Kenschaft, the first image that enters my mind is that of a Unicorn. Dr. Kenschaft is a unique person for which you would be hard pressed to find an equal. Her ability to teach mathematics to virtually anyone
Algebra and Trigonometry
Reviewed by "the_big_smooth", (New York)
This book comprehensively and completely covers all topics of advanced Algebra, and allows the reader to fully understand both the basics of Trigonometry, as well as enlighten him/her on some advanced topics in this field. It is well-writ
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MERA search of MERLOT learning exercisesCopyright 1997-2013 MERLOT. All rights reserved.Sat, 18 May 2013 21:08:42 PDTSat, 18 May 2013 21:08:42 PDTMER4434Finding the Domain of a Function Online Lesson
This lesson was created by Jennifer Anders and Nicole McGlashan of Huron High School, Huron, South Dakota. It is designed to help a student use the associated applet, and then extend the ideas it develops.
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Math Modeling
Math is definitely my favorite class even though my skills are somewhat lacking. Taught by Mr. Barys we learn math with a pretty interesting approach. If you're here to get a head start then it'll be a good idea to learn a programming language (we use a program called Mathematica ), but you'll get through it even if you don't know any programming. Mathematica is probably going to give you the biggest grief in the beginning of the year and if you can master this easily you're labs and POWs will become a lot easier.
What I like about math the most is how we take math that's actually challenging and we don't follow the "text book" approach. You do problems that make you think of math in different ways and it alters your perspective of math and how you view the world.
If you've never dealt with tetrahedrons start reading up. They're going to be with you for a while...
If you would like to see the kind of work we do, here are some of the very first labs done. All three utilized Mathematica:
GasWriteUp.pdf This was a small group porject that we worked on trying to model with trying to find how far should you go to find the best price. It was one of our first modeling things we've done so there are many things that we still have to learn, but hopefully it gives a taste on the kind of things we are trying to do.
Hanford,Oregon.pdf This lab was to learn how to use Mathematica to create linear models. Its basicly to get a handle and to become proficient in using the software.
TakeHome1.pdf This was the assesment that we had to do which sums up all our practices about linear models. The inclass part was a killer since you had no reference to other works (at least for me it was) but this one was a bit eaiser and I tried not to use my sources.
Composition Functions.pdf This assignment was to find out what happens to functions when certain functions are composed with each other. We were learning about the composition of functions which means using the output of one function as the input of another function: e.g f[x] and g[x] -> f[g[x]]. What we did before this lesson was discuss what the Tool box functions were (they're just the basic functions) and this lesson taught us how composing these functions together can make any function we wanted.
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Description and Objectives
Introduction to
use of computers to solve scientific and
engineering problems. Application
of mathematical judgment in
selecting tools to solve problems and to
communicate results.
MATLAB is the primary tool used for numerical
computation.
Although the subject matter of Beginning Scientific
Computing can be made rather difficult, I
will attempt to present the course
material in as simple a manner as
possible. More theoretical aspects, such
as proofs, will not be presented.
Applications will be emphasized.
Schedule and Homework
Follow links in
the table below to obtain a copy of the homework in Adobe
Acrobat (.pdf) format. You may also obtain here solutions to some of
the
homework and exam problems. An item shown below in plain text is not
yet
available. For additional information regarding viewing and printing
the
homework and solution sets,
click
here.
Grading
Your course grade will be calculated by weighing the homework,
the Midterm, and the Final in the proportions 50%, 20%, and 30%,
respectively. Homework problem
sets will be assigned bi-weekly. Homework
constitutes 50% of your final grade.
There will also be a one-hour-long
midterm and comprehensive final
for 20% and 30% or your grade respectively.
LATE HOMEWORK WILL NOT BE ACCEPTED.
Homework will be submitted and graded on-line. You have up to three
attempts per homework to get everything correct. If everything is correct
the first time a homework is submitted, you will receive a 100% for that homework.
If something is not correct, then you must fix it and re-submit the homework.
Your highest submitted homework grade will be your final grade for that particular
homework.
Matlab Resources
In this course, we will
make extensive use of Matlab, a technical computing
environment for
numerical computation and visualization produced by The MathWorks, Inc. A Matlab manual is
available in the MSCC Lab. If
you
are working in the Windows environment, be sure to check out the
Matlab
notebook feature that integrates Matlab with Microsoft Word.
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In this college level Calculus learning exercise, students use the ratio test to determine if a series converges or diverges. The one page learning exercise contains six problems. Solutions are not provided.
Students analyze geometric series in detail. They determine convergence and sum of geometric series, identify a series that satisfies the alternating series test and utilize a graphing handheld to approximate the sum of a series.
In this calculus learning exercise, students find the limit using the Limit Comparison Test and solve problems with series based on the p-series. They tell whether an equation will converge or diverge. There are 7 problems.
In this Calculus worksheet, students assess their understanding of various topics, including the derivatives of trigonometric functions, evaluating integrals, sigma notation, and convergent and divergent series. The one page interactive worksheet contains fifty-two problems. Answers are not provided.
Students investigate sequences and series numerically, graphically, and symbolically. In this sequences and series lesson, students use their Ti-89 to determine if a series is convergent. Students
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We bought it for our daughter and it seems to be helping her a whole bunch. It was a life saver. Margaret, CA
I really liked the ability to choose a particular transformation to perform, rather than blindly copying the solution process.. William Marks, OH
I never bought anything over the internet, until my neighbor showed me what the Algebra Buster can do, and I ordered one right away. It was a good decision! Thank you. Mark Fedor, MI07-06 :
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A video that focuses on the TI-Nspire graphing calculator in the context of teaching algebra. In this program the TI-Nspire is used to explore the nature of linear functions. Examples ranging from ... More: lessons, discussions, ratings, reviews,...
Zoom Algebra is a Computer Algebra System App for TI-83 Plus and TI-84 Plus graphing calculators. Its patent-pending interface is visual and easy to use, with many little shortcuts. For example, ...This lesson involves a useful applet from the National Library of Virtual Manipulatives (NLVM). It is a good introduction to solving two-step equations for grade 6 or 7 or an effective review lesson f... More: lessons, discussions, ratings, reviews,...
Student page for the classroom activity (also called the Masonry Problem; a variation on polyominoes) to be explored through manipulatives (dominoes). Students explore different possibilities of makin... More: lessons, discussions, ratings, reviews,...
Use this activity yourself or with students, to guide them through online math experiments. In the process, you'll get comfortable interpreting graphs
of time vs. distance, and gain insight into your... More: lessons, discussions, ratings, reviews,...
Gives students experience in manipulating graphs by changing domain and range values for the viewing window, which can easily be carried over to more powerful tools such as graphing calculators. Allow... More: lessons, discussions, ratings, reviews,...
A classroom activity (also called 1000 Lockers) to be explored through the use of manipulatives and a ClarisWorks spreadsheet. Students then look for patterns and write the answer algebraically. More: lessons, discussions, ratings, reviews,...
This eModule presents sequences of geometric patterns and encourages students to generate rules and functions describing relationships between the pattern number and characteristics of the pattern. S... More: lessons, discussions, ratings, reviews,...
A classroom activity, to be explored through large movement experience, manipulatives, and an interactive Java applet. Students then revisit the activity, look for patterns, and write the answer algeb... More: lessons, discussions, ratings, reviews,...
Compare different representations of motion: a story, a position graph, and the motion itself. Create a graph that matches a story, or write a story to match a graph, and check either by watching Mell... More: lessons, discussions, ratings, reviews,...
The primary goal of this lesson is to understand that the costs associated with buying on credit and that making only minimum payments are problematic to long-term financial health. The secondary goal... More: lessons, discussions, ratings, reviews,...
This collection of activities is intended to provide middle and high school Algebra I students with a set of data collection investigations that integrate mathematics and science and promote mathematiThis activity focuses on:
* graphing an ordered pair, (a, f(a)), for a function f
* the connection between a function, its table, and its graph
* the interpretation of the horizontal coordinate
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GCSE Maths: Creating and solving harder equations
Activities to enable learners of secondary mathematics to create and solve equations where the unknown appears more than once and to recognise that there may be more than one way of solving such equations. This is part of the 'Mostly algebra' set of materials from Standards unit: Improving learning in mathematics.
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Math made easier: advice from experts
Many students struggle with various kinds of math, including positive and negative number signs, fractions, factoring, graphing and word problems, instructors in the department of mathematics and statistics said.
In fall 2011, the success rate for college algebra, a core math course, was 59 percent, said Mellisa Hardeman, senior instructor in the department. The success rate dropped anther percentage point the following year, she said.
In fall 2012, 50 to 60 percent of pre-core math students had difficulties solving math problems, said Denise LeGrand, director of the Mac I math lab.
Ike McPhearson, math tutor, explained why students may have trouble comprehending math. One reason is that students may come from a home where education is not valued, he said.
A bad experience with an instructor can also change students' attitudes about math.
"You can't take yourself too seriously as a teacher," said Hardeman. Instructors can never give a student too much help passing math, she said.
Students who took a math course in high school before going to college are less likely to struggle with math, Hardeman said. Some students go to college years after graduating high school, however, and may forget everything they learned in their math classes.
Fortunately, there are a number of strategies that can help students overcome these challenges and develop a better understanding of math.
"In order to make math easy for students, show different ways of how to understand it," said McPherson, who has tutored high school and college students. Another way of making math fun for students is to create different games, he said.
According to LeGrand, the most important way to become better at math is to practice math exercises for 20 to 30 minutes.
"They won't see the results right away," said LeGrand, " but if they go to class and focus on work required, they will be successful and they will build confidence."
In addition, students can get help from tutors at the math lab. Each semester, the lab hires 12 tutors, LeGrand said.
For the math-impaired, there is a new math course called Quantitative and Mathematical Reasoning. The course was designed for students who are not science, technology, engineering or mathematics majors. It focuses on practical math, for example, currency exchange rates. The course fulfills the core math requirement, in place of college algebra.
Pre-core math courses, developmental math courses students take if they do not have the prerequisites for college math classes, are becoming more successful, said Tracy Watson, coordinator for pre-core math. The success rate for those courses rose to 77 percent in fall 2012, she said. Previously, the success rate was 37 percent for a 4-year period, she said.
This semester, there are 80 math majors at the university.
"We all like how math works because it all fits together," Watson said.
"Students who major in math develop a sense of thinking and solving problems," said Thomas McMillan, department chair.
Once students better understand math, they will have the confidence to solve not only math problems, but problems in everyday life as well
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mathematics to problem solving rather than derivation of theory. It provides a balance between physical and chemical hydrogeology. Numerous case studies cultivate student understanding of the occurrence and movement of ground water in a variety of geologic settings.
You can earn a 5% commission by selling Applied Hydrogeology
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Book Description: The Student Solutions Manual contains worked-out solutions to odd-numbered problems in the text. It displays the detailed process that students should use to work through the problems. The manual also provides interpretation of the answers and serves as a valuable learning tool for the student.
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Discrete Mathematics With Application - 3rd edition
Summary: Susanna Epp's DISCRETE MATHEMATICS, THIRD EDITION provides a clear introduction to discrete mathematics. Renowned for her lucid, accessible prose, Epp explains complex, abstract concepts with clarity and precision. This book presents not only the major themes of discrete mathematics, but also the reasoning that underlies mathematical thought. Students develop the ability to think abstractly as they study the ideas of logic and proof. While learning about such concept...show mores as logic circuits and computer addition, algorithm analysis, recursive thinking, computability, automata, cryptography, and combinatorics, students discover that the ideas of discrete mathematics underlie and are essential to the science and technology of the computer age. Overall, Epp's emphasis on reasoning provides students with a strong foundation for computer science and upper-level mathematics courses.
Benefits:
NEW!-Coverage of new topics suggested by the IEEE/CS and ACM joint committee on computing curriculum, including probability axioms and expected value, conditional probability and Bayes theorem, modular arithmetic, the Chinese remainder theorem, and RSA cryptography. Regular expressions and finite-state automata are also included, as recommended in by the ACM and IEEE-CS joint committee on the software engineering curriculum.
A flow-chart shows the prerequisite relations among the chapters, and most sections are divided into subsections so that instructors can easily tailor the book to meet the needs of their courses.
NEW!-Applications involving the Internet.
NEW!-Refinements to the exposition and exercises for improved pedagogy.
NEW!-Many new problems.
Every concept in the book is applied in at least one and often many different ways to motivate students. Eleven sections are explicitly devoted to applications to computer science, and additional applications are included in most sections.
A spiral approach, in which a number of concepts appear in increasingly more sophisticated forms, provides useful review and develops mathematical maturity in natural stages.
The book presents the unspoken logic and reasoning that underlie mathematical thought in a way that can be understood by typical freshman and sophomore college students.
In showing students how to discover and construct proofs and disproofs, Epp describes the kind of approaches that mathematicians use when confronting challenging problems in their own research.
A wealth of examples written in problem-solution form, and a large variety of exercises at all levels of difficulty are provided
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Book Description: This quick-reference dictionary for math students, teachers, engineers, and statisticians defines more than 700 terms related to algebra, geometry, analytic geometry, trigonometry, probability, statistics, logic, and calculus. It also lists and defines mathematical symbols, includes a brief table of integrals, and describes how to derive key theorems. Filled with illustrative diagrams and equations
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Hands-On Equations | Education
Profile Video
About Us
Hands-On Equations is a supplementary program that can be used with any math curriculum. It uses the visual and kinesthetic approach developed by Dr. Henry Borenson to provide students with an algebraic foundation for success with algebra. The program also provides students with a unique five-step procedure enabling them to concretize and solve word problems. Even ten lessons of Hands-On Equations can make a dramatic difference in the ability of your students to be successful with algebraic linear equations and word problems! Hands-On Equations provides a foundation for the Common Core State Standards and, in many instances, exceeds those standards.
Additional Information
What are the benefits of using Hands-On Equations?
No algebraic prerequisites are required
It is a game-like approach that fascinates and motivates students
The gestures or "legal moves" used to solve the equations reinforce the concepts at a deep kinesthetic level
The program can be used as early as the 3rd grade with gifted students, 4th grade with average students and 5th grade with LD students; it also serves as an excellent component of a middle-school pre-algebra program
Students attain a high level of success with the program
The program provides students with a strong foundation for later algebraic studies
The concepts and skills presented are essential for success in an Algebra 1 class
Algebra concepts your student will learn in only seven lessons!
The concept of an unknown
How to evaluate an expression
How to combine like terms
The relational meaning of the equal sign (both sides have the same value)
The meaning of an algebra equation
How to balance algebra equations (using the subtraction property of equality)
The concept of the check of an equation
The ability to solve one and two-step algebra equations
Solving equations with unknowns on both sides
How to work with a multiple of a parenthetical expression
But students learn much more; they learn that:
Mathematics is a subject one can understand
Mathematics can be learned without memorization
They need not be intimidated by algebra symbols
They can enjoy doing mathematics
They can explain mathematics to others
They can have success in one of the most "difficult" topics of mathematics
Services
Additional Links
Specials
Free on-site staff development with the purchase of thirty class sets for teacher and thirty students at $275 for each class set.!
Testimonials
"The Making Algebra Child's Play workshop is the most requested workshop from classroom teachers...not only is Hands-On Equations a powerful tool and the workshop exceptional, the customer service is outstanding." -Charity D. Weber, Specialist, Elementary MathematicsDistrict 8, Los Angeles School DistrictLos Angeles, CA "For the first time in my life, I actually understood how to do algebra and why it works. You and your child do not have to fear algebra. With Hands-On Equations, the solution to algebra is in your hands! I give it an A+!"- The Old SchoolHouse- The Magazine for Homeschooling Families
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Discovering Algebra: An Investigative Approach
*This is a good text for discovering Algebra 1 principles through investigation and discovery. The text is not overly wordy and makes sense in context with the examples and illustrations within each section.
*One significant improvement over most discovery learning texts and programs is the attention
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Introduction of the GMAT, the CAT methodology, GMAT strategic preparation plan, and use of the GMATWorkshop mistake log, GMAT Math I – GMATWorkshop Data sufficiency best practice; timing/pacing best practice and guessing strategies for time-pressure situations. Number properties; divisibility rules. GMATWorkshop Data sufficiency best practice.
Session 2:
Roots and powers, percentage and fractions, etc. All building-block concepts such as odds and evens, prime number, fractions, factorials and functions discussed with most representative examples.
Session 3:
Concepts and examples about Ratio, proportion and variation, statistics, mixture and alligations, speed, time and distance, races, etc.
Session 4:
An overview of critical concepts including necessary conditions, sufficient conditions and others. Different types of arguments and major/minor types of CR questions are discussed, such as typical methods to strengthen, weaken an argument, or find out the assumptions. Common logical fallacies and CR strategies are explained so that you can apply them in the Critical reasoning section and the analytical writing part.
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Mathematics for the Trades, CourseSmart eTextbook, 8th Edition
Description
For Basic Math, Math for the Trades, Occupational Math, and similar basic math skills courses servicing trade or technical programs at the undergraduate/graduate level.
THE leader in trades and occupational mathematics, Mathematics for the Trades: A Guided Approach focuses on fundamental concepts of arithmetic, algebra, geometry, and trigonometry. It supports these concepts with practical applications in a variety of technical and career vocations, including automotive, allied health, welding, plumbing, machine tool, carpentry, auto mechanics, HVAC, and many other fields. The workbook format of this text makes it appropriate for use in the traditional classroom as well as in self-paced or lab settings. For this revision, the authors have added over 150 new applications, new chapter summaries for quick review, and a new chapter on basic statistics. Student will find success in this clear and easy to follow format which provides immediate feedback for each step the student takes to ensure understanding and continued attention.
Table of Contents
Chapter 1 Arithmetic of Whole Numbers
Chapter 2 Fractions
Chapter 3 Decimal Numbers
Chapter 4 Ration, Proportion, and Percent
Chapter 5 Measurement
Chapter 6 Pre-Algebra
Chapter 7 Basic Algebra
Chapter 8 Pratical Plane Geometry
Chapter 9 Solid Figures
Chapter 10 Triangle Trigonometry
Chapter 11 Advanced Algebra
Chapter 12 Statistics
Answers to Previews
Answers
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A Look Ahead
A look ahead is a short weaving guide from the Nuffield Mathematics Project intended mainly for teachers of older students in upper primary and lower secondary.
The object of the guide was to re-state the aims and methods of the project in the light of experience so far, and to consider some of the problems which were then arising. It attempts to answer the question of 'where is it going?'. Other questions discussed are 'How does this fit in with CSE?' or even the 11-plus. Others have wondered 'Why new mathematics?' and whether what we've already written has any relevance to the outside world of today.
The contents list may help the reader to find where their own particular problems are discussed
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Abstract
Highlights the importance of involving students in patterns of generalized thinking to help them understand the underlying structure of arithmetic. Main areas of algebra in primary education; Discussion on the concept of arithmetic compensation; Models and materials that can be used to teach the compensation rule
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Overview - ELEMENTS CURRICULUM-BASIC GEOMETRY 10 STUDENT TEXTS
The Elements of Basic Geometry provide understanding of concepts in geometry for students with learning disabilities or reading at a lower level. The content matter is grade level, age-appropriate, and standards-aligned.
Students will learn to identify, describe, and analyze geometrical shapes. They will learn to describe and use mathematical relationships, compare and contrast lines, points, and planes with practical, real-life applications. The low learning level allows students to progress independently through each self-explanatory lesson.
Teacher's Edition includes reproducible practice worksheets, goals and objectives, chapter activities and projects, and two unit-test formats (standard form and form B for cognitively challenged students).
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Supermarket Math - Keith Devlin (Devlin's Angle)
Is it really possible to acquire skills - mathematical or otherwise - in a de-contextualized fashion in a classroom setting, then make use of those skills in any real-life situation that may require them? The assumption that this is indeed possible is
...more>>
Super Maths World - P. and R. May
Topic-based multiple choice assessment with a video-game feel. Year-long subscriptions available for schools/districts, with a reduced rate for small schools and individual customers. Some activities are free to guests.
...more>>
The Superstring Theory - Paul Hoiland
A site that gives a broad background from a mathematical point of view on general relativity, quantum mechanics, supersymmetry, and string theory. It also contains the concepts and math behind a registered modification to M-Theory that is non-perturbative.
...more>>
Supporting Cyber Students Over the Web - Philip Uys
Publications of Phillip Uys whose interests include open learning and distance education via the Internet and intranets ie networked education and its application. Papers include: "Supporting Cyber Students Over The Web: The On-line Campus of Massey University
...more>>
Surfaces Beyond the Third Dimension - Thomas Banchoff
A site that documents an art exhibit hosted by the Providence, Rhode Island Art Club in 1996, featuring computer-generated artwork designed by Tom Banchoff in collaboration with Davide Cervone (now at Union College) and student associates at Brown University.
...more>>
Susan C. Levine - The University of Chicago
Biography, courses, and more about Levine, the co-director of the Center for Early Childhood Research. Her freely downloadable articles include "Early puzzle play: A predictor of preschoolers' mental rotation skill"; "The role of parents and teachers
...more>>
SyllabusWeb - Syllabus Press, Inc.
From the publishers of Syllabus Magazine, a technology magazine for high schools, colleges, and universities. Highlights of recent issues of the magazine and full text archives of all Press publications. The June 1995 issue covers telecommunications andSymbolic Solutions Group - Erich Kaltofen
The Symbolic Solutions Group is a group of researchers in computer algebra and related subjects at North Carolina State University. Their articles are available for download, usually in PostScript format, via FTPetric Presentations - J. N. Bray
Presentations of groups, symmetric or otherwise, divided into classes to make location easier. Groups that have just one (finite) non-abelian composition factor up to isomorphism are listed in the first seven sections according to the isomorphism type
...more>>
Symmetry - Rick Engel
A lesson on the symmetry of the Platonic and Archimedean solids using Polymorf, a math manipulative product. Demonstrates inscription, truncation, sectioning, and space filling with polyhedra.Synergetics on the Web - Kirby Urner, 4D Solutions
R. Buckminster Fuller's "explorations in the geometry of thinking" is detailed in two thick volumes, Synergetics and Synergetics 2. Synergetics thoroughly permeates the rest of Fuller's writings as well. The invention for which Fuller is most famous,
...more>>
SysQuake - Calerga
Highly interactive software for the design and simulation of dynamic systems. Runs on PowerPC Macintosh computers and Windows 95, 98 and NT 4 computers. SysQuake LE is the free version of SysQuake, and may be downloaded from the site.
...more>>
SysteMATHics - Sigurd Andersen
Based on a discipline started by John G. Bennett, systematics is an attempt to understand the meaning implicit in number. The three areas of mathematical exploration covered are triadic numbers; n-grams, number patterns expressed graphically using the
...more>>
Systematic Mathematics - Paul Ziegler
This video-based home school math curriculum promises to "repair [the] essential foundation for your more experienced student, or create a solid math foundation for younger students." Each module consists of dozens of short lessons, delivered via DVD,
...more>>
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15 Free and Open Source Calculators for Students
"6. Calc+: An advanced calculator available for Android devices.
It includes many advanced scientific functions and a unit
conversion calculator.
"7. Calc Pro Free: Calc Pro Free comes with the standard and
scientific calculators found in the full version or you can build
your own calculator by purchasing the calculators and features you
need. Five calculator modes include simple, algebraic, direct
algebraic, expression and RPN. Apps are available for iPad,
iPhone/iPod, Windows 7 Phone, Windows Mobile and PC Desktop.
"8. Complex Numbers Calculator: Using the complex numbers
calculator the answers to algebra problems is only as far as your
mobile phone. All the basic operations for complex are provided:
addition, subtraction, multiplication, division, square root and
modulus. This calculator requires a Java J2ME MIDP 2.0 and CLDC 1.1
compatible mobile phone.
"9. Graphing Calculator: A graphical function plotting software
that can be used to represent any function of one variable, f(x).
The 2D function plotter provides an easy-to-use interface allowing
you to see the graph of a chosen function in a very short time.
This calculator requires a Java J2ME MIDP 2.0 and CLDC 1.1
compatible mobile phone.
"10. HandyCalc: This Android OS app is designed to be a symbolic
calculates system to handle some complicated mathematical features
including the capability to solve equation and equation set, define
variables, define your own functions, plot graph."
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Foundations of Math Diagnostic & Workshop
Are you wondering whether you are ready to learn – or relearn – GMAT math?
The vast majority of our students join a Manhattan GMAT course having not touched the math concepts tested on the exam in years. For some students, a review of the basics can be a useful refresher before diving into a full prep program. Manhattan GMAT offers Foundations of Math (FoM) workshops that are designed to reinforce fundamental math concepts and skills to help students succeed in our full-length courses. See whether our FoM workshops are right for you.
Assess Your Skills
with our Foundations of Math Basic Math Diagnostic
This 40 minute diagnostic, containing 20 problems at a 300-500 level of difficulty, will help determine how ready you are for a full-length course.
After completing the diagnostic, you will be given an assessment report indicating whether:
These two 3-hour workshops, taught by our expert Instructors, provide a strong understanding of fundamental concepts, processes, and skills.
FoM I: Covers Equations and Divisibility & Exponents
FoM II: Covers Fractions, Decimals, & Percents and Geometry
All workshop students will receive a copy of the Foundations of GMAT Math Strategy Supplement, and access to the accompanying online Homework Banks with hundreds of additional practice problems. All registered students will gain access to Manhattan GMAT's 6 CAT Exams as well.
Full-length course students and Guided Self-Study students receive the FoM workshops for free. If you purchase a FoM workshop and later sign up for a course or Guided Self-Study program, the cost of the workshop is deducted from your payment.
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Buy now
Short description Written by an avid user of topological groups, this book provides a concrete introduction to metric and topological groups. Readers learn how to replace the sequences with nets to obtain a general proof, and the result is an increased emphasis on topological groups and less on general topology. The exercises have been designed to help hold the interest of advanced and beginning readers, and a variety of calculations, remarks, and necessary related facts have been placed within the exercises and are used throughout the text.
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Math
Students must complete three credits math in grades 9 through 11. Most students take four or five credits of math, a significant number of whom reach Calculus or a higher level of math.
Department Philosophy
It is the belief of the Math Department that a qualified student who gives the appropriate amount of effort can find success in our program. We do not track students but encourage them to continually challenge themselves to become proficient in the subject area that best matches their ability. Thus a student in a regular class who meets the requirements may be promoted to an advanced class the following year, while a student who is not succeeding in the rigorous environment of an advanced or regular class may be moved into a B-level class.
The Math Department endorses and incorporates the National Council of Teachers of Mathematics standards and philosophies regarding math instruction. Through the course of a Westtown math education, a student will be exposed to many different ways of defining and framing problems, as well as solving them. Collaborative work, in-depth exploratory projects, and an emphasis on thinking skills pervade our curriculum as we seek more creative ways to help our students learn. Technology is used, depending on student needs and teacher interest, while still maintaining a rigorous grounding in writing clear, analytical mathematics. Traditional concerns about organizing students' work and knowledge motivate us to encourage the neatness, thoroughness, and clarity of thought and expression necessary for success in math and across the disciplines. The Math Department is also committed to furthering equity in its active encouragement of both male and female students, weaker and stronger students, as well as those from all cultural backgrounds. We believe that math is not just a cornerstone of intellectual development but also essential to effective participation as citizens in our democracy and in the world.
The goals of the mathematics curriculum:
To identify mathematical questions, to generalize from particular examples, and to use abstract reasoning
To analyze data, represent it graphically, and gain experience creating mathematical models for the systems generating the data
To expose students to a rigorous, theoretical development of math systems both algebraic and geometric
To gain experience in using calculators, programming languages and applications software
Distinguishing Features:
Flexibility of the curriculum: many paths and different paces are possible.
Curriculum is able to meet a wide variety of student interests and background preparation, very responsive to individual student's needs. We challenge all students at an appropriate level
The graphing calculator has been integrated as an educational tool.
Faculty stay abreast of and respond to the latest developments in research, pedagogy, technology, and curriculum.
Math Lab is staffed by current teachers offering help daily in the Learning Resource Center.
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Calculus and Analysis
In mathematics , catastrophe theory is a branch of bifurcation theory in the study of dynamical systems ; it is also a particular special case of more general singularity theory in geometry . Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation.
In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos . It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter
A means of describing how one state develops into another state over the course of time. Technically, a dynamical system is a smooth action of the reals or the integers on another object (usually a manifold ). When the reals are acting, the system is called a continuous dynamical system, and when the integers are acting, the system is called a discrete dynamical system.
Disclaimer: CosmoLearning is promoting these educational resources as a courtesy of Gaussian Department of Mathematics (GMath).
Visualizing a function can give a mathematician enormous insight into the function's algebraic and geometrical properties. The easiest way to see what a function looks like is to use a computer as a graphing tool.
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This was easily one of my favorite books as a kid (I took Calculus as a sophomore in high school, Multi-Variable Calculus as a junior, and Linear Algebra/Differential Equations as a senior). And now, after many years of no math, it looks like I'm going to be tutoring students in high-school level mathematics.
So where am I going? Not to the standard Algebra text, or any massive textbook that they use in schools. I'm going to Gullberg. Glad I kept it. ( )
What does mathematics mean? Is it numbers or arithmetic, proofs or equations? Jan Gullberg starts his massive historical overview with some insight into why human beings find it necessary to "reckon," or count, and what math means to us. From there to the last chapter, on differential equations, is a very long, but surprisingly engrossing journey. Mathematics covers how symbolic logic fits into cultures around the world, and gives fascinating biographical tidbits on mathematicians from Archimedes to Wiles. It's a big book, copiously illustrated with goofy little line drawings and cartoon reprints. But the real appeal (at least for math buffs) lies in the scads of problems--with solutions--illustrating the concepts. It really invites readers to sit down with a cup of tea, pencil and paper, and (ahem) a calculator and start solving. Remember the first time you "got it" in math class? With Mathematics you can recapture that bliss, and maybe learn something new, too. Everyone from schoolkids to professors (and maybe even die-hard mathphobes) can find something useful, informative, or entertaining here. --Therese Littleton
(retrieved from Amazon Sat, 05 Jan 2013 15:58:43 -0500)
▾Library descriptions
An illustrated exploration of mathematics and its history, beginning with a study of numbers and their symbols, and continuing with a broad survey that includes consideration of algebra, geometry, hyperbolic functions, fractals, and many other mathematical functions.… (more)
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The second course in a three part calculus sequence. Topics include: the Riemann integral, applications of integration, techniques of integration, and transcendental functions. Prerequisite: MATH 215 with grade of C or higher.
Prerequisite(s) / Corequisite(s):
MATH 215 with a grade of C or higher.
Text(s):
Most current editions of the following:
Most current editions of the following:
Calculus
By Finney, R. & G. Thomas. (Addison-Wesley) Recommended
Calculus
By Stewart (Brookes-Cole) Recommended
Course Objectives
To use calculus to formulate and solve problems and communicate solutions to others.
To use technology as an integral part of the process of formulation, solution and communication.
To understand and appreciate the connections between mathematics and other disciplines.
Measurable Learning Outcomes:
• Compute definite integrals as the limit of Riemann sums and approximate integrals using finite Riemann sums. • Evaluate definite and indefinite integrals using the Fundamental Theorem of Calculus and the method of substitution. • Compute areas and volumes using definite integrals. • Identify the natural exponential and logarithmic functions as inverses of each other and find their derivatives and integrals. • Solve exponential growth and decay problems arising from biology, physics, chemistry, and other sciences. • Compute derivatives and integrals of functions containing inverse trigonometric functions. • Analyze various indeterminate forms and apply L'Hospital's rule to evaluate limits of such forms. • Use the Substitution Rule and the Integration by Parts formula to evaluate indefinite and definite integrals. *Describe and explain special methods required to integrate trigonometric and rational functions. * Apply numerical methods of integration such as Simpson's Rule to approximate definite integrals.
Topical Outline:
Integrals
Applications of integrals
Transcendental functions
Techniques of integration
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Event Description
In this beginning Sketchpad webinar, you'll see how Sketchpad makes it simple and intuitive for students to graph functions, explore families of functions, and transform the functions they graph. You'll learn how to use the various graphing and function commands so you can start using them in your classroom tomorrow! In addition, we'll explore how to use ready-made demonstration sketches to incorporate Sketchpad into your precalculus or advanced algebra class.
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Choose a format:
Used - Paperback
Overview
Book Details
Painless Geometry
English
Series:
Painless Ser.
ISBN:
0764142305
EAN:
9780764142307
Category:
Juvenile Nonfiction / Mathematics / Geometry
Publisher:
Barrons Educational Series, Incorporated
Release Date:
08/04/2012
Synopsis:
The author demonstrates how solving geometric problems amounts to fitting parts together to solve interesting puzzles. Students discover relationships that exist between parallel and perpendicular lines; analyze the characteristics of distinct shapes such as circles, quadrilaterals, and triangles; and learn how geometric principles can solve real-world problems. Titles in Barrons Painless Seriesare written especially for middle school and high school students who are having a difficult time with a specific subject. In many cases, a student is confused by the subjects complexity and details. Still other students simply finds a subject uninteresting, an attitude that usually results in lower grades. Painlesstitles offer informal, student-friendly approaches to each subject, emphasizing interesting details, supplementing the text with amusing insights, and outlining potential pitfalls clearly and step by step. Students begin to understand how disparate details all fit together to form a clear picture. Timelines, ideas for interesting projects, and Brain Tickler quizzes in many of these titles help to take the pain out of study and improve each students grades.
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Mathematics
As the world and the economy become more reliant on technology, the need for people with strong mathematical backgrounds will only increase. Mathematicians have the ability to think through problems in a logical and rigorous manner, to see patterns in data, and to formulate models and strategies to solve problems. Mathematicians are employed in almost every sector of the economy. In finance, they help create financial instruments and determine the amount of risk in investments. In manufacturing, they design quality control procedures and streamline processes. In medicine, they design models to see the effects of drugs and to help decide the proper dosages.
The Mathematics program at Martin Methodist gives each student a wide base of knowledge to start their mathematical careers. Leaving the program, students should be ready for graduate school, an entry-level job in mathematics, or a job as a teacher. Because the Mathematics program at Martin Methodist is small, we are able to give every student personal attention. The Special Topics course allows us to offer majors a course that piques their interests. The Senior Project gives students an opportunity to explore some topic of their own choosing, expanding their knowledge and view of mathematics.
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Request Information
MATH201: Intermediate College Algebra
Course Credits:
3
Course Hours Per Week:
12
Course Overview
This course introduces intermediate college algebra concepts and their applications through problem-solving and the use of equations. A graphing calculator (Texas Instruments TI-83) is required to complete homework assignments. This course provides a basis for college algebra, trigonometry, and other higher-level mathematics courses.
Students will examine such key topics as:
• Linear equations and inequalities
• Systems of linear equations
• Exponents and polynomials
• Rational expressions and functions
• Radicals and rational exponents
• Quadratic equations, functions, and inequalities
Students will complete sets of problems using an online homework manager, MyMathLab. Students will also participate in discussions designed to help them apply concepts to word problems and daily life. Students will complete both a mid-term and a final comprehensive exam.
Course Learning Objectives
Apply elementary algebra to solve mathematical problems.
Use functions, graph linear equations and inequalities.
Explain systems of equations in two and three variables.
Comprehend and use inequalities and absolute value.
Make use of polynomials, rational expressions, exponents and radicals.
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Numbers, Groups and Codes
9780521540506
ISBN:
052154050X
Edition: 2 Pub Date: 2004 Publisher: Cambridge Univ Pr
Summary: A thoroughly revised and updated version of the popular textbook on abstract algebra. The material is introduced with clarity and reference to problems and concepts that students will easily understand. With many examples and exercises, it will serve as the ideal introduction to this important and ubiquitous subject.
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Sets and their representation; Union, intersection and complement of sets and their algebraic properties; Power set; Relation, Types of relations, equivalence relations, functions;. one-one, into and onto functions, composition of functions.
UNIT 2 : COMPLEX NUMBERS AND QUADRATIC EQUATIONS:
Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, algebra of complex numbers, modulus and argument (or amplitude) of a
complex number, square root of a complex number, triangle inequality, Quadratic equations in real and complex number system and their solutions. Relation between roots and co-efficients, nature of roots, formation of quadratic equations with given roots.
UNIT 3 : MATRICES AND DETERMINANTS:
Matrices, algebra of matrices, types of matrices, determinants and matrices of order two and three. Properties of determinants, evaluation of determinants, area of triangles using determinants. Adjoint and evaluation of inverse of a square matrix using determinants and elementary transformations, Test of consistency and solution of simultaneous linear equations in two or three variables using determinants and matrices.
UNIT 4 : PERMUTATIONS AND COMBINATIONS:
Fundamental principle of counting, permutation as an arrangement and combination as selection, Meaning of P (n,r) and C (n,r), simple applications.
UNIT 5 : MATHEMATICAL INDUCTION:
Principle of Mathematical Induction and its simple applications.
UNIT 6 : BINOMIAL THEOREM AND ITS SIMPLE APPLICATIONS:
Binomial theorem for a positive integral index, general term and middle term, properties of Binomial coefficients and simple applications.
Evaluation of simple integrals of the type Integral as limit of a sum. Fundamental Theorem of Calculus. Properties of definite integrals. Evaluation of definite integrals, determining areas of the regions bounded by simple curves in standard form.
UNIT 10: DIFFERENTIAL EQUATIONS:
Ordinary differential equations, their order and degree. Formation of differential equations. Solution of differential equations by the method of separation of variables, solution of homogeneous and linear differential equations of the type: dy+ p (x) y = q (x) dx
UNIT 11: CO-ORDINATE GEOMETRY:
Cartesian system of rectangular co-ordinates 10 in a plane, distance formula, section formula, locus and its equation, translation of axes, slope of a line, parallel and perpendicular lines, intercepts of a line on the coordinate axes. Straight lines Various forms of equations of a line, intersection of lines, angles between two lines, conditions for concurrence of three lines, distance of a point from a line, equations of internal and external bisectors of angles between two lines, coordinates of centroid, orthocentre and circumcentre of a triangle, equation of family of lines passing through the point of intersection of two lines. Circles, conic sections
Standard form of equation of a circle, general form of the equation of a circle, its radius and centre, equation of a circle when the end points of a diameter are given, points of intersection of a line and a circle with the centre at the origin and condition for a line to be tangent to a circle, equation of the tangent. Sections of cones, equations of conic sections (parabola, ellipse and hyperbola) in standard forms, condition for y = mx + c to be a tangent and point (s) of tangency.
UNIT 12: THREE DIMENSIONAL GEOMETRY:
Coordinates of a point in space, distance between two points, section formula, direction ratios and direction cosines, angle between two intersecting lines.Skew lines, the shortest distance between them and its equation. Equations of a line and a plane in different forms, intersection of a line and a plane, coplanar lines.
UNIT 13: VECTOR ALGEBRA:
Vectors and scalars, addition of vectors, components of a vector in two dimensions and three dimensional space, scalar and vector products, scalar and vector triple product.
UNIT 14: STATISTICS AND PROBABILITY:
Measures of Dispersion: Calculation of mean, median, mode of grouped and ungrouped data calculation of standard deviation, variance and mean deviation for grouped and ungrouped data. Probability: Probability of an event, addition and multiplication theorems of probability, Baye's theorem, probability distribution of a random variate, Bernoulli trials and Binomial distribution.
Force and Inertia, Newton's First Law of motion; Momentum, Newton's Second Law of motion; Impulse; Newton's Third Law of motion. Law of conservation of linear momentum and its applications, Equilibrium of concurrent forces. Static and Kinetic friction, laws of friction, rolling friction. Dynamics of uniform circular motion: Centripetal force and its applications.
UNIT 4: WORK, ENERGY AND POWER
Work done by a constant force and a variable force; kinetic and potential energies, workenergy theorem, power. Potential energy of a spring, conservation of mechanical energy, conservative and nonconservative forces; Elastic and inelastic collisions in one and two dimensions.
UNIT 5: ROTATIONAL MOTION
Centre of mass of a two-particle system, Centre of mass of a rigid body; Basic concepts of rotational motion; moment of a force, torque, angular momentum, conservation of angular momentum and its applications; moment of inertia, radius of gyration. Values of moments of inertia for simple geometrical objects, parallel and perpendicular axes theorems and their applications. Rigid body rotation, equations of rotational motion.
UNIT 6: GRAVITATION
The universal law of gravitation. Acceleration due to gravity and its variation with altitude and depth. Kepler's laws of planetary motion. Gravitational potential energy; gravitational potential. Escape velocity. Orbital velocity of a
Thermal equilibrium, zeroth law of thermodynamics, concept of temperature. Heat, work and internal energy. First law of thermodynamics. Second law of thermodynamics: reversible and irreversible processes. Carnot engine and its efficiency.
UNIT 9: KINETIC THEORY OF GASES
Equation of state of a perfect gas, work doneon compressing a gas.Kinetic theory of gases - assumptions, concept of pressure. Kinetic energy and temperature: rms speed of gas molecules; Degrees of freedom, Law of
Electric charges: Conservation of charge, Coulomb's law-forces between two point charges, forces between multiple charges; superposition principle and continuous charge distribution. Electric field: Electric field due to a point charge, Electric field lines, Electric dipole, Electric field due to a dipole, Torque on a dipole in a uniform electric field. Electric flux, Gauss's law and its applications to find field due to infinitely long uniformly charged straight wire, uniformly charged infinite plane sheet and uniformly charged thin spherical shell. Electric potential and its calculation for a point charge, electric dipole and system of charges; Equipotential surfaces,
Electrical potential energy of a system of two point charges in an electrostatic field. Conductors and insulators, Dielectrics and electric polarization, capacitor, combination of capacitors in series and in parallel, capacitance of a parallel plate capacitor with and without dielectric medium between the plates, Energy stored in a capacitor.
UNIT 12: CURRRENT ELECTRICITY
Electric current, Drift velocity, Ohm's law, Electrical resistance, Resistances of different materials, V-I characteristics of Ohmic and nonohmic conductors, Electrical energy and power, Electrical resistivity, Colour code for resistors; Series and parallel combinations of resistors; Temperature dependence of resistance. Electric Cell and its Internal resistance, potential difference and emf of a cel l, combination of cells in series and in paral lel. Kirchhoff's laws and their applications. Wheatstone bridge, Metre bridge. Potentiometer - principle and its applicat ions.
UNIT 13: MAGNETIC EFFECTS OF CURRENT AND MAGNETISM
Biot - Savart law and its application to current carrying circular loop. Ampere's law and its applications to infinitely long current carrying straight wire and solenoid. Force on a moving charge in uniform magnetic and electric fields.Cyclotron. Force on a current-carrying conductor in a uniform magnetic field. Force between two parallel current-carrying conductors-definition of ampere. Torque experienced by a current loop in uniform magnetic field; Moving coil galvanometer, its current sensitivity and conversion to ammeter and voltmeter. Current loop as a magnetic dipole and its magnetic dipole moment. Bar magnet as an equivalent solenoid, magnetic field lines; Earth's magnetic field and magnetic elements. Para-, dia- and ferro- magnetic substances. Magnetic susceptibility and permeability, Hysteresis, Electromagnets and permanent magnets.
Reflection and refraction of light at plane and spherical surfaces, mirror formula, Total internal reflection and its applications, Deviation and Dispersion of light by a prism, Lens Formula, Magnification, Power of a Lens, Combination of thin lenses in contact, Microscope and Astronomical Telescope (reflecting and refracting) and their magnifyingpowers. Wave optics: wavefront and Huygens' principle, Laws of reflection and refraction using Huygen's principle. Interference, Young's double slit experiment and expression for fringe width. Diffraction due to a single slit, width of central maximum. Resolving power of microscopes and astronomical telescopes, Polarisation, plane polarized light; Brewster's law, uses of plane polarized light and Polaroids.
Propagation of electromagnetic waves in the atmosphere; Sky and space wave propagation, Need for modulation, Amplitude and Frequency Modulation,Bandwidth of signals, Bandwidth of Transmission medium, Basic Elements of a Communication System (Block Diagram only).
SECTION –B
UNIT 21: EXPERIMENTAL SKILLS
Familiarity with the basic approach and observations of the experiments and activities: and time.
4. Metre Scale - mass of a given object by principle of moments.
5. Young's body liquid by method of mixtures.
11. Resistivity of the material of a given wire using metre bridge.
12. Resistance of a given wire using Ohm's law.
13. Potentiometer –
(i) Comparison of emf of two primary cells.
(ii) Determination of internal resistance of a cell.
14. Resistance and figure of merit of a galvanometer by half deflection method.
15. Focal length of:
(i) Convex mirror
(ii) Concave mirror, and
(iii) Convex lens
using parallax method.
16. Plot of angle of deviation vs angle of incidence for a triangular prism.
17. Refractive index of a glass slab using a travelling microscope.
18. Characteristic curves of a p-n junction diode in forward and reverse bias.
19. Characteristic curves of a Zener diode and finding reverse break down voltage.
20. Characteristic curves of a transistor and finding current gain and voltage gain.
Fundamentals of thermodynamics: System and surroundings, extensive andintensivesublimation, phase transition, hydration, ionization and solution.Second law ofthermodynamics; Spontaneity of processes; DS of the universe and DG of the system as criteria for spontaneity, Dgo (Standard Gibbs energychange) and equilibrium constant.
specific.
UNIT 9 : CHEMICAL KINETICS
Rate of a chemical reaction, factors affecting the rate of reactions:concentration, temperature, pressure and catalyst; elementary and complex reactions, order and molecularity of reactions, rate law, rate constant and its
units, differential and integral forms of zero and first order reactions, their characteristics and half - lives, effect of temperature on rate of reactions – Arrhenius theory, activation energy and its calculation, collision theory of
Modes of occurrence of elements in nature, minerals, ores; Steps involved in the extraction of metals - concentration, reduction (chemical and electrolytic methods) and refining with special reference to the extraction of Al, Cu, Zn and Fe; Thermodynamic and electrochemical principles involved in the extraction of metals.
UNIT 13: HYDROGEN
Position of hydrogen in periodic table, isotopes, preparation, properties and uses of hydrogen; Physical and chemical properties of water and heavy water; Structure, preparation, reactions and uses of hydrogen peroxide; Hydrogen as a fuel.
UNIT 14: S - BLOCK ELEMENTS (ALKALI AND ALKALINE EARTH METALS)
Group - 1 and 2 Elements
General introduction, electronic configuration and general trends in physical and chemical properties of elements, anomalous properties of the first element of each group, diagonal relationships. Preparation and properties of some important compounds - sodium carbonate and sodium hydroxide; Industrial uses of lime, limestone, Plaster of Paris and cement; Biological significance of Na, K, Mg and Ca.
UNIT 15: P - BLOCK ELEMENTS
Group - 13 to Group 18 Elements
General Introduction: Electronic configuration and general trends in physical andchemical properties of elements across the periods and down the groups; unique behaviour of the first element in each group. Groupwise study of the p – block elements
General methods of preparation, properties, reactions and uses. Amines: Nomenclature, classification, structure, basic character and identification of primary, secondary and tertiary amines and their basic character.
Diazonium Salts: Importance in synthetic organic chemistry.
UNIT 25: POLYMERS
General introduction and classification of polymers, general methods of polymerization-addition and condensation, copolymerization; Natural and synthetic rubber and vulcanization; some important polymers with emphasis on their monomers and uses - polythene, nylon, polyester and bakelite.
Part - I Awareness of persons, places, Buildings, Materials.) Objects, Texture related to Architecture and build~environment. Visualising three dimensional objects from two dimensional drawings. Visualising. different sides of three dimensional objects. Analytical Reasoning Mental Ability (Visual, Numerical and Verbal).
Part - tre es, plants etc.) and rural life.
Note: Candidates are advised to bring pencils, own geometry box set, erasers and colour pencils and crayons for the Aptitude Test.
Haryana Public Service Commission invites applications from eligible candidates for to the following 151 Administrative and Executive posts :
1. HCS (Executive Branch) : 30 posts
2. Dy. S.P. : 09 posts
3. E.T.O. : 38 posts
4. District Food and Supplies Controller: 01 post
5. Tehsildar 'A' Class :16 posts
6. Assistant Registrar Co-Operative Society:08 posts
7. Assistant Excise & Taxation Officer : 05 posts
8. Block Development and Panchayat Officer : 17 posts
9. Traffic Manager: 03 posts
10. District Food & Supplies Officer :03 posts
11. Assistant Employment Officer: 21 posts
How 27/12/2011. (Last date is 03/01/2011 for the candidates of far-flung areas)
Monday, 28 November 2011
Online/ Offline application are invitedfor Himachal Pradesh, whose applications are received by post from these areas is 10/01/2011 :
How to Apply : Apply Online at HPPSC website on or before 26/12/2011 or Application in the prescribedOMR application form should be sendto the Controller of Examinations, Himachal PradeshPublic Service Commission, Nigam Vihar, Shimla-2, on or before26/12/2011.
How to Apply : Application in prescribed format should be sent in an envelope superscribed with bold letters as "Application for the posts of .................... " on or before 16/12/2011 (23/12/2011 for candidates from far-flung areas) toOffice of the Regional Director, Staff Selection Commission, (Western Region) 1ST Floor, Pratishtha Bhavan, 101, MK Road, Mumbai - 400020.
Application Fee: Rs.500/- (Rs.50/- for SC/ST/PWD), should be paid at any branch of the Syndicate Bank in the prescribed payment challan. Keep original counterfoil of the challan with you as it is to be produced at the time of written test along with call letter.
How to Apply: Apply online at Syndicate Bank website only from 25/11/2011 to 15/12/2011.Take a printout for future references as this is to be submitted at the later stage
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How is that you can walk into a classroom and gain an overall sense of thequality of math instruction taking place there? What contributes to gettingthat sense? In Math Sense, Chris Moynihan explores some of the componentsthat comprise the look, sound, and feel of effective teaching and learning.Does the landscape of the classroom feature such items... more...
This book (along with vol. 2)... more...
A collection of lectures presented at the Sixth International Conference, held at the University of Ioannina, on p-adic functional analysis with applications in the fields of physics, differential equations, number theory, probability theory, dynamical systems, and algebraic number fields. more...
While computational technologies are transforming the professional practice of mathematics, as yet they have had little impact on school mathematics. This pioneering text develops a theorized analysis of why this is and what can be done to address it. It examines the particular case of symbolic calculators (equipped with computer algebra systems) in... more...
The Hindu?Arabic numeral system (1, 2, 3,...) is one of mankind's
greatest achievements and one of its most commonly used
inventions. How did it originate? Those who have written about the
numeral system have hypothesized that it originated in India; however,
there is little evidence to support this claim.
This book provides considerable evidence... more...
This text presents an up-to-date treatment of fuzzy automata theory and fuzzy languages. The authors also discuss applications in a variety of fields, including databases, medicine, learning systems and pattern recognition. more...
Thirty-three papers from the July 2003 conference on non-associative algebra held in Mexico present recent results in non-associative rings and algebras, quasigroups and loops, and their application to differential geometry and relativity. The contributors investigate alternating triple systems with simple Lie algebras of derivations, simple decomp more...
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College Geometry : A Discovery Approach - Text Only - 2nd edition
Summary: College Geometry is an approachable text, covering both Euclidean and Non-Euclidean geometry. This text is directed at the one semester course at the college level, for both pure mathematics majors and prospective teachers. A primary focus is on student participation, which is promoted in two ways: (1) Each section of the book contains one or two units, called Moments for Discovery, that use drawing, computational, or reasoning experiments to guide students to an oft...show moreen surprising conclusion related to section concepts; and (2) More than 650 problems were carefully designed to maintain student interest.
Features NEW! Geometer's Sketchpad Projects. A part of each exercise section, and incorporated into selected examples. NEW! Glossary. End of Chapter Material. In addition to the current chapter summary and End of Chapter True/False questions, there are new conceptual exercises to test the students' understanding of the chapter material. Moments for Discovery. Reinforces chapter material by encouraging students to experiment. Historical perspective. Appropriate biographies are written throughout the text, to give context to the material that students are learning. Our Geometric World. Placed throughout each chapter, this feature illustrates the real world application of the material that students are learning. ...show less
2000-11-25 Paperback Very Good Very Good! Used texts may contain bookstore stickers on cover. Used texts may NOT contain supplemental materials such as CD's, info-trac, access codes, etc...Satisfac...show moretion Guaranteed! Choose Expedited Shipping for fastest delivery! Free USPS Tracking Number. NO INTERNATIONAL ORDERS. ...show less
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This study presents the results of the evaluation of middle grade mathematics curriculum from China, Russia, and the United States. The study focused on the algebraic concept of inequalities and the geometric concept of angles formed by the intersection of a transversal with non-parallel and parallel lines.
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Mathematics Support Materials
This page has been optimised for Internet Explorer 5.0 or above. If you are accessing it using Netscape or any other browser then please click here.
About This Project:
The aim of this project is to produce a library of portable, interactive, web based support packages to help students learn various mathematical ideas and techniques and to support classroom teaching. The packages are in Adobe's Portable Document Format (PDF). These packages will introduce the mathematical ideas and their rules. The packages will use the linking capabilities of PDF files to generate exercises and quizzes and so allow the students to test their understanding of the material with immediate feedback. We will also use our backgrounds in science to construct smaller support packages to show applications of mathematics in science and engineering.
To view the files, you will need the latest (free) version of Acrobat Reader: Acrobat Reader 6.0. Detailed instructions about how to use the packages can be found here.
The Packages:
Note that the packages may be either worked through on line or downloaded and used on any computer. They are compact enough to fit on a floppy disk.
The Software:
Since HTML, the language of the web, is poorly suited to mathematical formulae, this project uses LaTeX and PDF. LaTeX is a variant of TeX, which is the world standard for typesetting mathematics. It produces professional quality output which is universally accepted by scientific publishing companies. We use LaTeX to produce PDF (portable document format) output which can be put on the web. These PDF files are read using the freely available and widely distributed Adobe Acrobat Reader.
These packages were produced using LaTeX. This was then converted into PDF files. The LaTeX code makes much use of various packages which have been developed by D.P. Story. Some information about TeX and LaTeX can be found here.
Acknowledgements:
We are grateful to David McMullan for discussions on many aspects of this work and thank Arsen Khvedelidze for his help. We thank HEFCE for funding our work as part of the PPLATOFDTL4 project and our colleagues in PPLATO for useful discussions. We are also grateful to the Higher Education Academy Engineering Subject Centre for funding us via a Mini-Project grant
in collaboration with Frank Hamer and LTSN Physical Sciences for a development project grant, in collaboration with Simon Belt, and for the opportunity to present these ideas at various workshops.
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Exam
1
This will cover Chapter 0 (excluding propositional logic and truth
tables), Chapter 1 and Chapter 2.
The best study guide for this exam are the first four homework
assignments. You will be asked to prove things.
Homework#2, due September 3
Chapter 0, Problems #6, 7, 14, 15, 16
And these problems:
A. Prove that if a number's last digit is "8," then it is even
B. For what values of n
is the sum 1+2+3+ ... + n an
even number? Prove your answer.
Tuesday
August 25
Topics Covered
Introduction to Proofs, definitions of odd and even,
proof that the product of two odd numbers is odd
proof that an A by B rectangle it tileable with 1x2 dominos if and only
if at least one of A, B is even.
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Specification
Aims
To give an introduction to Lebesgue measure on R. To show how this theory leads to the Lebesgue integral on R, and to introduce the concept of Hausdorff dimension of sets in Rn.
Brief Description of the unit
Riemann created the integral named after him (familiar to students of MT2222) in 1854. The great achievement of the Riemann integral is that it integrates any continuous function defined on a bounded interval in R. But the Riemann theory has the deficiency that it does not behave well with respect to taking limits.
In 1902, in his thesis, H. Lebesgue introduced the integral that bears his name. His key idea was to extend the notion of length from intervals to more complicated sebsets of R (and Rn). His integral integrates any function which is Riemann integrable, and also has good limit properties.
In this course unit we will give a modern treatment of Lebesgue's theory. Although much of the theory can now be done in much more generality than was the case in Lebesgue's time, this course will be focused on the real line setting. The material is proof oriented and should appeal to students who have successfully taken MATH20222 (Real Analysis). However, MATH20222 is not a prerequisite for the course, and any ideas required that are not in first or second year core course units will be reviewed when needed.
The course unit should be useful to students taking probability course units in years three and four since the ideas of measure and integral have a central role in probability theory.
Learning Outcomes
On successful completion of the course unit the students will be able to
understand how Lebesgue measure on R is constructed.
understand the notions of measurable functions on R
know how Lebesgue integration is derived from Lebesque measure
know how to manipulate integrals and use the basic theorems
understand the notion of Hausdorff dimension of sets in Rn, and be able to calculate it for simple examples.
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Algebra by Design
Second Edition
by Russell F. Jacobs
This popular publication contains 44 one-page activities which provide review and practice of basic algebraic concepts and skills. The second edition, published in June of 2012, contains four new exercises on opposites and finding the missing term in a sequence.
Click here to download the table of contents in pdf format. Click here to download a sample activity in pdf format.
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Mathematics
Statement of Goals and Objectives
The goals of the department are to offer a complete high-school mathematics curriculum for the college-bound student and to challenge each individual to develop her God-given mathematical talents.
Objectives
To develop logical and creative approaches to problem solving.
To develop facility in applying basic mathematical concepts.
To stimulate clarity and precision in language usage.
To encourage an appreciation for the deductive nature of mathematics.
To guide the student in selecting courses that allow her maximum achievement for her abilities, needs and interests.
To insure a smooth transition to mathematics courses at the college level.
Requirements
Four credits in mathematics are required for graduation, which must include one semester of trigonometry. A Texas Instruments graphing calculator is required for all mathematics classes. The model numbers of the calculators that may be used are announced in the spring.
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Cheat Sheets and Tables. Here is list of cheat sheets and tables that Iu0027ve written. Most of these are pdf files and so you will need the Adobe Viewer to view them.
tutorial.math.lamar.edu/../cheat_table.aspx
AQA GCSE Specification, 2010 - Mathematics A 5 Background Information 1 Introduction Following a review of the National Curriculum requirements, and the establishment of the ...
store.aqa.org.uk/../AQA-4306-W-SP-10.PDF
Page 5 Introduction The GCSE awarding bodies have prepared revised specifications to incorporate the range of features required by GCSE and subject criteria.
AQA GCSE Specification, 2011 - Mathematics A 5 Background Information 1 Introduction Following a review of the National Curriculum requirements, and the establishment of the ...
store.aqa.org.uk/../AQA-4306-W-SP-11.PDF
23 Areas outside the box will not be scanned for marking 25 The table shows information about two types of light bulbs, Standard and Energy Saving. Both types of light bulb give ...
Dear Mathematics Colleague We are proud to introduce you to the new accredited GCSE Mathematicsspecification from Edexcel, ready for first teaching in September 2010.
B_Issue 2_WEB.pdf
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Math circles are periodic math programs that attract middle and high school students to mathematics by exposing them to intriguing and intellectually stimulating topics, rarely encountered in classrooms. The concept of a "math circle" has been modeled after experiences in Russia, Bulgaria, Romania, and other countries. It is a way to stimulate, encourage and help gifted and interested pre-college students to study and solve mathematics problems, sometime with the involvement of mathematicians and/or mathematics faculty from universities. The main purpose of a math circle is to inspire in students an understanding of and a lifelong love for mathematics. The book under review presents materials used during the course of one year in a math circle organized by mathematics faculty at Moscow State University, and also used at the mathematics magnet school known as Moscow School Number 57. This volume contains 28 Problems Sets. Each problem set has a similar structure: it combines review material with a new topic, offering problems in a range of difficulty levels. The problems are nonstandard; solving them takes wit, thinking outside the box, and, sometimes, hours of contemplation. Some are within the reach of most mathematically competent high school students, while others are difficult even for a mathematics professor. Many mathematically inclined students have found that tackling these problems, or even just reading their solutions, is a great way to develop mathematical insight. The problems and the accompanying material are well suited for math circles. They are also appropriate for problem-solving classes and practice for regional and national mathematics competitions. In summary, this is an excellent resource for those interested in math circles, including students and parents (they can just skip the organizational part and go directly to the presentations and problems). For those interested in starting and running a math circle, I think it is an invaluable resource.
Reviewer:
Vicenţiu D. Rădulescu (Craiova)
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MAS202 Advanced Calculus
This course continues the study of the calculus of functions of two variables, begun in MAS170. It includes the application of partial
derivatives to finding and classifying local maxima and minima. The new concept of a line integral is introduced, and related to double integrals via Green's theorem, a kind of two-dimensional Fundamental Theorem of Calculus. Returning to functions of a single variable, the important techniques of Fourier series and Fourier transforms are introduced, with a taste of some of their many applications.
Economics dual students will get a handout on Lagrange multipliers at the beginning.
1. Line integrals.
Review of integration as a limit of sums. Definition of line
integral as a limit of sums. Work done as motivating example. Basic
properties. Calculating line integrals using parametrised curves.
Integrals of exact differentials, finding the potential function.
Criterion for exactness, path-independence. (3 lectures)
6. Maxima and minima. Review of critical points for
functions of one variable. Connection with Taylor series.
Critical points for functions of n variables, recipe for their
classification when n=2. Taylor series for functions of n
variables. Quadratic forms and classification of critical points.
Constrained maxima and minima, Lagrange multipliers. Use of the
MAPLE commands mtaylor, Eigenvalues and extrema. (6 1/2 lectures)
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MAT 165 Contemporary Mathematics
Use a variety of problem solving techniques, including inductive and deductive reasoning.
Make investigations of mathematical ideas and be able to use patterns and observations to make conjectures.
Question the reasonableness of a solution in relation to the original problem.
Approximate mental calculations and develop estimation skills.
Demonstrate a basic understanding about the nature of numbers used in mathematics and have an appreciation of our numeration system in terms of some historical numeration systems. Demonstrate a facility in using the operations of the real numbers and an understanding of the properties of real numbers.
Use variables to represent mathematical quantities and expressions; represent mathematical functions and relationships using tables, graphs and equations.
Demonstrate an understanding of basic geometric concepts such as parallelism, perpendicularity, congruence, similarity and symmetry.
Objectives
Recognize and use a variety of problem solving techniques that include inductive and deductive reasoning.
Demonstrate problem-solving techniques with a variety of problems including financial management, probability and statistics.
Demonstrate an understanding of the nature of numbers and develop an appreciation of our numeration system including its properties and operations.
Demonstrate an understanding of basic geometric and trigonometric concepts and apply those concepts to solve problems.
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...EngCalc is a new calculating software with extended capabilities. This is an essential application...and engineering student. The main advantage of the software is a simple input format even for the...to navigate through. The compact size of the software does not hinder the performance. To the contrary,...
...Free problem-solving mathematics software allows you to work through more than 500...of linear, quadratic, biquadratic, reciprocal, cubic and fractional algebraic identities, equations and inequalities, solutions of trigonometric and...hyperbolic equations and more. The software supports a number of interface styles. This program...
...purpose of OpenDDPT project is to build a software that can aids people to build automatically these...control devices. The strategy used by this software is based on the concept that differential equations...
...with complex equations. This modern and beautiful scientific software makes the exchange of complex ideas simple.
The...scientific equation editor used in this software allows you to create complex letters, including complex...with all kinds of brackets
+ Symbols of algebra and math analysis
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...ESBPDF Analysis is Probability Analysis Software that provides everything needed for using Discrete and...Excel) give P(X less than A) and using algebra other results can be found whereas ESBPDF Analysis...is Probability Analysis Software that handles all the combinations for you.
Features...
...Looking for a software utility that would help math students plot various...it very practical for solving in-class and homework algebra or calculus problems. The program comes with customizable...is much easier to use than most similar software titles. Program's user interface is well labeled and...
...This bilingual problem-solving mathematics software allows you to work through 36319 arithmetic and...students to review and reinforce their skills. The software supports bilingual interface and a number of interface...
...This bilingual problem-solving mathematics software allows you to work through 5018 algebra equations...and repetition rather than through rote memorization. The software includes linear, quadratic, biquadratic, reciprocal, cubic and fractional...students to review and reinforce their skills. The software supports bilingual interface and a number of interface...
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Discrete Mathematical Structures, Sixth Edition, offers a clear and concise presentation of the fundamental concepts of discrete mathematics. Ideal for a one-semester introductory course, this text contains more genuine computer science applications than any other text in the field. This b...
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Feynman's Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics
This new volume contains four previously unpublished lectures that Feynman gave to students preparing for exams. With characteristic flair, insight and humor, Feynman discusses topics students struggle with and offers valuable tips on solving physics problems. An illuminating memoir by Matthew Sands — who originally conceived The Feynman Lectures on Physics— gives a fascinating insight into the history of Feynman's lecture series and the books that followed. This book is rounded off by relevant exercises and answers by R. B. Leighton and R. E. Vogt, originally developed to accompany the Lectures on Physics.
It is important that schools emphasize a problem-solving approach to mathematics beginning in the early years and continuing through high school. Students should learn to value the process of solving ...
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Math and Physics - Articles
Wed, 22 May 2013 09:52:48 +000086400Resources for mathematics and physics that relate to game development, including collisions and algorithmsMath for Game Developers: Intro to Matrices
Note:
This is an ongoing series of videos that will be updated every week. When a new video is posted we will update the publishing date of this article and the new video will be found at the end of the playlistAdvanced Vectors
]]>Tue, 30 Apr 2013 05:19:59 +0000571df53e6b6fc55ba413313a42bb4cf5Math for Game Developers: Intro to Vectors
Starting with the second series, "Advanced Vectors", you can download the source code that I'm using from GitHub, from the description of each video. If you have questions about the topics covered or requests for future topics, I would love to hear them! Leave a comment, or ask me on my Twitter, @Intro to Vectors
]]>Sun, 21 Apr 2013 13:03:25 +000057947ed4d4130c7ff0a057c8654dd1a3Vectors and Matrices: A Primer
Preface
This article is designed for those who need to brush up on your maths. Here we will discuss vectors, the operations we can perform on them, and why we find them so useful. We'll then move onto what matrices and determinants are, and how we can use them to help us solve systems of equations. Finally, we'll move onto using matrices to define transformations in space.
Note:
This article was originally published to GameDev.net back in 2002. It was revised by the original author in 2008 and published in the book Beginning Game Programming: A GameDev.net Collection, which is one of 4 books collecting both popular GameDev.net articls and new original content in print format.
Vectors
Vector Basics – What is a vector?
Vectors are the backbone of games. They are the foundation of graphics, physics modelling, and a number of other things. Vectors can be of any dimension, but are most commonly seen in two, three, or four dimensions. They essentially represent a direction, and a magnitude. Thus, consider the velocity of a ball in a football game. It will have a direction (where it's travelling), and a magnitude (the speed at which it is travelling). Normal numbers (i.e. single dimensional numbers) are called scalars however I'll only use two here: vector equations and column vectors.
Vectors can be written in terms of its starting and ending position, using the two end points with an arrow above them. So, if you have a vector between the two points A and B, you can write that as:
A vector equation takes the form:
a = xi + yj + zk
The coefficients of the i, j, and k parts of the equation are the vectors components. These are how long each vector is in each of the 3 axis.
For example, the vector equation pointing to the point ( 3, 2, 5 ) from the origin ( 0, 0, 0 ) in 3D space would be:
a = 2i + 3j + 5k
The second way I will represent vectors is as column vectors. These are vectors written in the following form:
Where x, y, and z are the components of that vector in the respective directions. These are exactly the same as the respective components of the vector equation. Thus in column vector form, the previous example could be written as:
There are various advantages to both of the above forms, although column vectors will continue to be used. Various mathematic texts may use the vector equation form.
Vector Mathematics
There are many ways in which you can operate on vectors, including scalar multiplication, addition, scalar product, vector product and modulus.
Modulus
The modulus or magnitude of a vector is simply its length. This can easily be found using Pythagorean Theorem with the vector components. The modulus is written like so:
a = |a|
Given:
Then,
Where x, y and z are the components of the vector in the respective axis.
Addition
Vector addition is rather simple. You just add the individual components together. For instance, given:
The addition of these vectors would be:
This can be represented very easily in a diagram, for example: The individual components are simply subtracted from each other. The geometric representation however is quite different from addition. For example:
The visual representation
It may be easier to think of this as a vector addition, where instead of having:
c = a – b
We have:
c = -b + a
Which according to what was said about the addition of vectors would produce:
You can see that putting a on the end of –b has the same result.
Scalar Multiplication
This is another simple operation; all you need to do is multiply each component by that scalar. For example, let us suggest that you have a vector a and a scalar k. To perform a scalar multiplication you would multiply each component of the vector by that scalar, thus
The Scalar Product (Dot Product)
The scalar product, also known as the dot product, is very useful in 3D graphics applications. The scalar product is written:
This is read "a dot b".
The definition of the scalar product is:
Θ is the angle between the two vectors a and b. This produces a scalar result, hence the name scalar product. This operation has the result of giving the length of the projection of a on b. For example:
The length of the thick gray horizontal line segment would be the dot product.
The scalar product can also be written in terms of Cartesian components as:
We can put the two dot product equations equal to each other to yield:
With this, we can find angles between vectors.
Scalar products are used extensively in the graphics pipeline to see if triangles are facing towards or away from the viewer, whether they are in the current view (known as frustum culling), and other forms of culling.
The Vector Product (Cross Product)
The vector product, also commonly known as the cross product, is one of the more complex operations performed on vectors. In simple terms, the vector product produces a vector that is perpendicular to the vectors having the operation applied. Great for finding normal vectors to surfaces!
I'm not going to get into the derivation of the vector product here, but in expanded form it is:
Read "a cross b".
Since the cross product finds the perpendicular vector, we can say that:
i x j = k
j x k = i
k x i = j
Note that the resultant vectors are perpendicular in accordance with the "right hand screw rule". That is, if you make your thumb, index and middle fingers perpendicular, the cross product of your middle finger with your thumb will produce your index finger.c = a x b
This first finds the vector perpendicular to the plane made by a and b then scales that vector so it has a magnitude of 1.
One important point about the vector product is that:
This is a very important point. If you put the inputs the wrong way round then you will not get the correct normal.
Unit Vectors
These are vectors that have a unit length, i.e. a modulus of one. The i, j and k vectors are examples of unit vectors aligned to the respective axis. You should now be able to recognise that vector equations are quite simply just that. Adding together 3 vectors scaled by varying degrees to produce a single resultant vector.
To find the unit vector of another vector, we use the modulus operator and scalar multiplication like so:
For example:
That is the unit vector b in the direction of a.
Position Vectors
These are the only type of vectors that have a position to speak of. They take their starting point as the origin of the coordinate system in which they are defined. Thus, they can be used to represent points in that space, and v is the unit vector giving its direction. t is called the parameter and scales v. From this you can see that as t varies a line is formed in the direction of v.
This equation is called the parametric form of a straight line. Using this to find the vector equation of a line through two points is easy:
If t is constrained to values between 0 and 1, then we have a line segment starting at the point p0 and p1.
Using the vector equation we can define planes and test for intersections. A plane can be defined as a point on the plane, and two vectors that are parallel to the plane.
Where s and t are the parameters, and u and v are the vectors that are parallel to the plane. Using this, you can find the intersection of a line and a plane, as the point of intersection must line on both the plane at the line. Thus, we simply make the two equations equal to each other.
Given the line and plane:
To find the intersection we equate so that:
We then solve for w, s and t, and plug them into either the line or plane equation to find the point. When testing for a line segment w must be in the range 0 to 1.
Another representation of a plane is the normal-distance. This combines the normal of the plane, and its distance from the origin along that normal. This is especially useful for finding out what sides of a plane points are. For example, given the plane p and point a:
p = n + d
Where,
The point a is in front of the plane p if:
This is used extensively in various culling mechanisms.
Matrices
What is a Matrix anyway?
A matrix can be considered a 2D array of numbers, and take the form:
Matrices are very powerful, and form the basis of all modern computer graphics. We define a matrix with an upper-case bold type letter, as shown above:
The identity matrix can be any dimension, as long as it is also a square matrix.
The elements of a matrix are all the numbers within it. They are numbered by the row/column position such that:
The zero matrix is one that has all its elements set to 0.
Vectors can also be used in column or row matrices. I will use column matrices here, as that is what I have been using in the previous section. A 3D vector a in matrix form will use a matrix A with dimension 3x1 so that:
Which as you can see is the same layout as using column vectors.
Matrix Arithmetic
I'm not going to go into every possible matrix manipulation (we would be here some time), instead I'll focus on the important ones.
Scalar / Matrix Multiplication
To perform this operation all you need to do is simply multiply each element by the scalar. Thus, given matrix A and scalar k:
Matrix / Matrix Multiplication
So generally, given three matrices A, B and C, where C is the product of A and B. A and B have dimension mxn and pxq respectively. They are conformable if n=p. The matrix C has dimension mxq. It is said that the two matrices are conformable if their inner dimensions are equal (n and p here).
The multiplication is performed by multiplying each row in A by each column in B. Given:
So, with that in mind let us try an example!
It's as simple as that! Some things to note:
A matrix multiplied by the identity matrix is the same, so:
AI = IA = A
The Transpose
The transpose of a matrix is it flipped along the diagonal from the top left to the bottom right and is denoted by using a superscript T, for example:
Determinants
Determinants are a useful tool for solving certain types of equations, and are used rather extensively.
Let's take a 2x2 matrix A:
The determinant of matrix A is written |A| and is defined to be:
That is the top left to bottom right diagonal multiplied together subtracting the top right to bottom left diagonal. Things get a bit more complicated with higher dimensional determinants, let us discuss a 3x3 determinant first. Take A as:
Step 1: move to the first value in the top row, a11. Take out the row and column that intersects with that value.
Step 2: multiply that determinant by a11.
We repeat this along the top row, with the sign in front of the result of step 2 alternating between a "+" and a "-". Given this, the determinant of A becomes:
Now, how do we use these for equation solving? Good question.
Given a pair of simultaneous equations with two unknowns:
We first push these coefficients of the variables into a determinant, producing:
You can see that it is laid out in the same way. To solve the equation in terms of x, we replace the x coefficients in the determinant with the constants k1 and k2, dividing the result by the original determinant:
To solve for y we replace the y coefficients with the constants instead. This algorithm is called Cramers Rule.
Let's try an example to see this working, given the equations:
We push the coefficients into a determinant and solve:
To find x substitute the constants into the x coefficients, and divide by D:
To find y substitute the constants into the y coefficients, and divide by D:
It's as simple as that! For good measure, let's do an example using 3 unknowns in 3 equations:
Solve for x:
Solve for y:
Solve for z:
And there we have it, how to solve a series of simultaneous equations using determinants, something that can be very useful.
Matrix Inversion
Equations can also be solved by inverting a matrix. Using the same equations as before:
We first push these into three matrices to solve:
Let's give these names such that:
We need to solve for B (this contains the unknowns after all). Since there is no "matrix divide" operation, we need to invert A and multiply it by D such that:
Now we need to know how to actually do the matrix inversion. There are many ways to do this, and the way that I'm going to use here, multiplied by the following expression:
Where i and j is the position in the matrix.
For example, given a 3x3 matrix A, and its co-factor C. To calculate the fist element in the cofactor matrix (c11), we first need to get rid of the row and column that intersects this so that:
c11 would then take the value of the following:
We would then repeat for all elements in matrix A to build up the co-factor matrix C. The inverse of matrix A can then be calculated using the following formula.
The transpose of the co-factor matrix is also referred to as the adjoint.
Given the previous example and equations, let's find the inverse matrix of A.
Firstly, the co-factor matrix C would be:
|A| is:
|A| = -2
Thus, the inverse of A is:
We can then solve the equations by using:
We can find the values of x, y and z by pulling them out of the resultant matrix, such that:
x = -62
y = 39
z = 3
Which is exactly what we got by using Cramer's rule!
Matrices are said to be orthogonal if its transpose equals its inverse, which can be a useful property to quick inverting of matrices.
Matrix Transformations
Graphics APIs use a set of matrices to define transformations in space. A transformation is a change, be it translation, rotation, or whatever. Using position vector in a column a matrix to define a point in space, a vertex, we can define matrices that alter that point in some way.
Transformation Matrices
Most graphics APIs use three different types of primary transformations. These are translation; scaling; and rotation. We can transform a point p using a transformation matrix T to a point p' like so:
p' = Tp
We use 4 dimensional vectors from now on, of the form:
We then use 4x4 transformation matrices. The reason for the 4th component here is to help us perform translations using matrix multiplication. These are called homogeneous coordinates. I won't go into their full derivation here, as that is quite beyond the scope of this article (their true meaning and purpose comes from points in projective space).
Translation
To translate a point onto another point, there needs to be a vector of movement, so that:
Where p' is the translated point, p is the original point and v is the vector along which the translation has taken place.
By keeping the w component of the vector as 1, we can represent this transformation in matrix form as:
Scaling
You can scale a vertex by multiplying it by a scalar value, such that:
Where k is the scalar constant. You can multiply each component of p by a different constant. This will make it so you can scale each axis by a different amount.
In matrix form this becomes:
Where kx, ky, and kz are the scaling factors in the respective axis.
Rotation
Rotation is a more complex transformation, so I'll give a more thorough derivation for this than I have the others.
Rotation in a plane (i.e. in 2D) can be described in the following diagram:
This diagram shows that we want to rotate some point p by ω degrees to point p'. From this we can deduce the following equations:
We are dealing with rotations about the origin, thus the following can be said:
|P'| = |P|
Using the trigonometric identities for the sum of angles:
We can expand the previous equations to:
From looking at the diagram, you can also see that:
Substituting those into our equations, we end up with:
Which is what we want (the second point as a function of the first point).
We can then push this into matrix form:
Here, we have the rotation matrix for rotating a point in the x-y plane. We can expand this into 3D by having three different rotation matrices, one for rotating along the x axis, one in the y axis, and another for the z axis (this one is effectively what we have just done). The unused axis in each rotation remains unchanged. These rotation matrices become:
Any rotation about an axis by θ can be undone by a successive rotation by –θ, thus:
Also, notice that the cosine terms are always on the top left to bottom right diagonal, and notice the symmetry of the sine terms along this axis. This means, we can also say:
Rotation matrices that act upon the origin are orthogonal.
One important point to consider is the order of rotations (and transformations in general). A rotation along the x axis followed by a rotation along the y axis is not the same as if it were applied in reverse. Similarly, a translation followed by a rotation does not produce the same result as a rotation followed by a translation.
Frames
A frame can be considered a local coordinate system, or frame of reference. That is, a set of three basis vectors (unit vectors perpendicular to each other), plus a position, relative to some other frame. So, given the basis vectors and position:
That is a, b, and c defines the basis vectors, with p being its position.
We can push this into a matrix, defining the frame, like so:
This matrix is useful, as it lets us transform points into a second frame – so long as those points exist in the same frame as the second frame. Thus, consider a point in some frame:
Assuming the frame we defined above is in the same frame as that point, we can transform this point to be in our frame like so:
That is:
Which if you think about it is what you would expect. If you have a point at -1 along the x axis, and you transform it into a frame that is at +1 along the x axis and orientated the same, then relative to that second frame the point appears at -2 along its x axis.
This is useful for transforming vertices between different frames, and is incredibly useful for having objects moving relative to one frame, while that frame itself is moving relative to another one. You can simply multiply a vertex first by its parent frame, then by its parents frame, and so forth to eventually get its position in world space, or the global frame.
You can also use the standard transformation matrices previously to transform frames with ease. For instance, we can transform the above frame to rotate by 90º along the y axis like so:
This is exactly what you would expect (the z axis to move to where the x axis was, and the x axis point in the opposite direction to the original z axis).
Summary
Well that's it for this article. We've gone through what vectors are, the operations that we can perform on them and why we find them useful. We then moved onto matrices, how they help us solve sets of equations, including how to use determinants. We then moved on to how to use matrices to help us perform transformations in space, and how we can represent that space as a matrix.
I hope you've found this article useful!
References
Interactive Computer Graphics – A Top Down Approach with OpenGL – Edward Angel
Primitive description
Let's start with a mathematical description of primitives we're going to intersect. For this article I've used just ray (line), plane, sphere, axis-aligned bounding box and triangle ... these should be enough for a basic game engine.
Ray
This is the simplest (and very useful) primitive you can think about. Let us define it with two 4-component numbers, point (further referred as origin) and direction. Mathematically the definition is:
Where:
x represents any point on a line
o represents some exact specified point on line (e.g. origin)
d represents direction vector of line
t is parameter within some specified range
The range specifies whether our line is infinite in both directions (+d and -d), whether it's just in a single direction, or whether it's finite.
Writing a ray object in some programming language is straight-forward of course, you should end up with something like this:
Constructor and methods are straight-forward (and should be also inline for performance reasons).
Plane
Another basic primitive. There are more ways to define a plane, but we will derive one that is well known and used. What is a plane? It's a point set that contains every such point in space, that when we create vector from that point to our known point on a plane, that vector has to be perpendicular to plane normal. And we know that perpendicular vectors are those whose dot product is equal to zero. So:
Where:
n is unit vector in the plane normal direction
x is every point on the plane
p is our specified point on the plane
Expanding this yields:
Substitute:
And we get something very familiar:
Which is a plane equation (so we proved that the definition we use is actually a plane). Definition in programming language could be like this:
Sphere
Yet another basic primitive, that is widely used. The definition is surface containing all points in distance from sphere's center less-or-equal to sphere's radius. We will define it using an analytic equation:
Axis-aligned Box
From here on I'll call this one just AABB (which stands for Axis-Aligned Bounding Box). Definition is very simple, as we work in Cartesian coordinate system, we have perpendicular axes X, Y and Z - AABB is defined as volume between minimum and maximum point in this system, the sides are made by planes orthogonal to planes formed by all 3 combinations of 2 base axis (XY, XZ and YZ). Mathematically we can define it as:
Triangle
The first and only primitive in this article that is going to be defined by more than 2 values, yet very important. Most of the game worlds are described by triangles, most of the simulation use objects described as triangles, and every more (or less) complex 3D object can be decomposed into triangles. They're awesome; as opposed to other boring N-gons, triangles are never concave!
Let's define triangle with 3 points, then any point on triangle can be described as:
With conditions:
Where:
x is every point in triangle
A, B, C are points defining triangle
a, ß, ? are so called barycentric coordinates (e.g. parameters in the equation above)
A triangle inside of a program could be defined as:
class Triangle
{
public:
float4 p[3];
<<constructor>>
<<methods>>
};
Intersections
So now that we have defined some of the basic primitives, it's time to derive intersection equation and also intersection algorithms between the primitives. As I want to keep the article short (and I don't want to storm you with zillion of equations and exhausting equation solving), I will just show you 4 derivations - Ray-Plane (analytic derivation), Ray-Sphere (geometric derivation), Ray-AABB (tricky derivation) and Ray-Triangle (more complex analytic derivation). You should get an idea of how the intersection equation can be derived and in the end you should be able to derive the rest of them on your own.
Ray-Plane intersection
I will derive Ray-Plane intersection analytically, as this is one of the most simple approaches to the derivation. When we want to derive some intersection algorithm analytically, we should write equations of those 2 objects right away, so (I intentionally used commonly known plane equation, instead of our definition):
Now we're looking for a point that lies on both, the ray and the plane - so we're looking for x. So we first write the first equation in scalar form:
Let's substitute these 3 to plane equation:
It's time to reverse substitution we did when "deriving" plane equation, e.g.:
And transformed to vector form we get:
But we can optimize this a little bit:
Voila, that's what we have been looking for! So basically we need just subtraction, 2 dot products and 1 division. In program this can look like (assuming we want to intersect planes in front of ray origin in direction of ray):
Ray-Sphere intersection
I could perform analytic derivation like in the Ray-Plane intersection derivation, although Ray-Sphere intersection is just one of the cases that can be easily derived using geometry. Let's start with an image:
We're now looking for 2 distances:
Finding C-O is quite straightforward, it's just simple:
Another thing we need to find is distance along the ray to point X, but as we know ray direction, it's just simple projection of L.
Right now we should look again at the image, we can see 3 important right angle triangle, one formed by {d, OX and OC}, another one {d, r, PX} and last one {d, r, P'X}. We need to know distance PX (respectively P'X, they're equal). So let's use Pythagorean theorem twice:
Just a side note. Analytic derivation in this case is also very trivial, you will end up with quadratic equation, where determinant is negative in case of no hit, zero in case of single point hit (e.g. tangent line), and positive in case of secant.
Let's jump ahead to boxes!
Ray-AABB intersection
Another important one, although a bit tricky. Let's start with an image again:
You can see, that box in 2D is formed by 2 slabs, horizontal and vertical. 3D is analogous to this, but of course it's formed by 3 slabs. For each slab we want to find minimum distance and maximum distance, e.g.:
Where a and b represents minimum (respectively maximum) point of our AABB. What we get is the minimum and maximum distance to each slab. Now we have to correctly order the distances along the ray direction, this is quite simple:
Where min/max is defined as:
Entry/exit point is then defined as maximum of near distances (see the image), thus:
Where hmax/hmin function is horizontal minimum/maximum, defined as:
Of course a hit only occurs when the exit point is larger than zero, and the exit point is further than the entry point (if it isn't, we missed the AABB). Time to code:
It can be a little more optimized, although I'll leave that to reader as an exercise.
Ray-Triangle intersection
I'd like to say, there is like a dozen of ways to intersect a ray with a triangle. Lots of them don't even use a triangle defined by 3 points, and re-define it in better suited way. I'll show you how to derive plain good old Barycentric test with one huge advantage, you don't have to precompute a triangle in any way, just use the standard definition with 3 points.
Let's start with equations we already have:
The triangle equation actually is equal to this:
And let's arrange it like this:
Substituting line equation for x yields:
Those are 3 equations with 3 unknown variables. Instead of thinking about some more clever way, let's solve it in brute force manner with Cramer's rule. First let's re-arrange it to better form:
For clarity let's substitute e1 for B-A, e2 for C-A and p for o-A.
Now it's time to apply Cramer's rule, e.g.:
But matrix determinant can be written as scalar triple product, so:
I know this might look scary right now, but it really is simple. For those that understand code easier than math I also add code:
Conclusion
Intersection algorithms, as I've said, are core algorithms for practically every game engine! And not just game engines, ray-tracers, physics simulators, collision detection libraries, even some AI simulations use them, etc. If you made it here, then I have to say thanks for reading my article, and I hope you've learned something new.
9 Apr 2013: Initial release]]>Tue, 09 Apr 2013 14:59:51 +0000527490f08486bf8af2b8d0bf6e73911bPractical use of Vector Math in Games
For a beginner, geometry in 3D space may seem a bit daunting. 2D on paper was hard enough, but now, geometry in 3D?
Good news: use of trigonometry in graphics is rare and avoided for multitude of reasons. We have other tools which are easier to understand and use. You may recognize our old friend here - a vector.
This article will introduce you to 3D vectors and will walk you through several real-world usage examples. Even though it focuses on 3D, most things explained here also work for 2D.
Article assumes familiarity with algebra and geometry, some programming language, a basic knowledge of OOP.
Vector Math in Games
Concepts
In mathematics, a vector is a construct that represents both a direction as well as a magnitude. In game development it often can be used to describe a change in position, and can be added or subtracted to other vectors. You would usually find a vector object as part of some math or physics library.
They typically contain one or more components such as x, y and z. Vectors can be 1D (contain only x), 2D (contain x, y), 3D (contain x, y, z), even 4D. 4D vectors can be used to describe something else, for example a color with an extra alpha value.
One of the toughest things beginners have difficulty with when it comes to vectors is to understand how something that seemingly looks like a point in space can be used to describe a direction.
Take the 2D vector (3,3). To understand how this represents a direction you need only look at the following graph. We all know that it takes two points to form a line. So what is the second point? The missing point is the origin located at (0,0). If we draw a line from the origin at (0,0) to (3,3) we get the following:
As you can see, the introduction of the origin as the second point gives our vector a direction. But you can also see that the first point (3,3) can be moved (or translated) closer to or farther away from the origin.
The distance from the origin is known as the magnitude and is given by the quadratic equation a^2 + b^2 = c^2. In this case 3^2 + 3^2 = c^2 and c = sqrt(18) ~= 4.24. If we divide each of the components of this vector by 4.24 we can scale the vector back to a point where it has a magnitude of just 1. We will learn in upcoming examples how this process of normalizing the vector can be very useful since this process retains the direction, but gives us an ability to scale the vector up or down quickly with simple multiplication by some numeric (aka scalar) value.
For the upcoming examples, I am going to assume that your math library uses Vector2 for a 2D vector and Vector3 for a 3D vector. There are various differences in naming across libaries and programming languages, for example vector, vector3, Vector3f, vec3, point, point3f and similar. Your math library must have some documentation and examples for them.
Note: In the world of programming programmers have utilized the vector type to represent both vectors in the traditional mathematical/physics sense as well as points or arbitrary n-tuplet units at their own discretion. Be advised.
Like any other variable, the vector in your code has a meaning which is up to you: it can be a position, direction, velocity.
Position - a vector represents an offset from your world origin point (0, 0, 0).
Direction - vector looks very much like an arrow pointing at some direction. That is indeed what it can be used for. For example, if you have a vector that points south, you can make all your units go south.
A special case of direction vector is a vector of length 1. It is called a normalized vector, or normal for short.
A velocity vector can describe a movement. In that case, it is a difference in position over specific amount of time.
Remember the Basics - Vector Addition and Subtraction
Vector addition is used to accumulate the difference described by both vectors into the final vector. For example, if an object moves by A vector, and then by B vector, the result is the same as if it would have moved by C vector.
For subtraction, the second vector is simply inverted and added to the first one.
Example: a Distance Between Objects
If your vectors represent positions of objects A and B, then B - A will be a difference vector between positions. The result would represent a direction and a distance A must travel to get to B.
For example, to get a distance vector to that tree you need to subtract your position from the tree position.
I am using pseudo-code to keep the code clean. Three numbers in parentheses (x, y, z) represent a vector.
Concept: Simulation
Wait! We do not want to update object once per second. In fact, we want to do it as often as possible. But we can not expect the time between updates to be fixed. So we use a delta time, which is the time since the last update.
Since our delta time represents a fraction of the time passed, we can use it to get only that fraction of our velocity for the next position update.
position += velocity * delta
This is a very basic simulation. To make one, we model how our objects behave in our world (cannon ball has constant velocity forever). Then we load initial game state (cannon ball starts with initial velocity and position).
The final piece where everything comes together is an update loop, which is executed regularly We use the delta time (time interval) since the previous update, and update every simulated object to be where it should on the rules we defined (update cannon ball position based on its velocity).
Example: Gravity, Air Resistance and Wind
Our cannon ball is quite boring: it will move to the same direction and at the same speed forever. We need it to react to the world around. For example, let there be gravity for it to fall, air resistance for it to slow down, and the wind just for kicks.
What the gravity actually means in a game? Well, it has a side effect of increasing velocity of objects to one direction: down. So in case our Y axis is up, our gravity vector would be:
# increase velocity of every object -2 down per second
gravity_vector = (0, -2, 0)
The main point of this example is to demonstrate how easy it is to create quite complex behavior using a simple vector math.
Concept: Direction
Often you will not need a distance from A to B, just the direction for A to face the B. A distance vector B - A can represent the direction, but what if you need to move "just a bit" towards B, at the exact speed you like?
In such case vector length should be irrelevant If we reduce a direction vector to the length of 1, we can use it for this, and other purposes. We call this reduction the normalization and the resulting vector the normal vector.
So, a normal vector always should be the length of 1, otherwise it is not a normal vector. A normal vector represents an angle without any actual "angles" required as a possible change in position. If we multiply the vector by a scalar number, we get the direction vector that has the same length as the scalar.
There should be a "normalize" function in your math library to get a normal vector from any other vector.
So we can get a movement by exactly the 3 units towards B.
final_change = (B - A).normalize() * 3
Concept: Surface Plane
A normal vector can also be used to describe a direction a surface plane is facing. You can imagine the plane as an infinite slice of the world in half at a particular point P. Rotation of this slice is defined by the normal vector N.
To rotate this slice/plane, you would change its normal vector.
Concept: Dot Product
A dot product is an operation on two vectors, which returns a number. You can think of this number as a way to compare the two vectors.
Usually written as:
result = A dot B
This comparison is particularly useful between two normal vectors, because it represents a difference in rotation between them.
If dot product is 1, normals face the same direction.
If dot product is 0, normals are perpendicular.
If dot product is -1, normals face opposite directions.
Here are the normal values in the illustration:
Note that change from 1 to 0 and from 0 to -1 is not linear, but follows a cosine curve. So, to get an angle from a dot product, you need to calculate arc-cosine of it:
angle = acos(A dot B)
Example: Light
Suppose we are writing a light shader and we need to calculate the brightness of a pixel at a particular surface point. We have:
A normal vector which represent the direction of surface at this point.
Position of the light.
Position of this surface point.
We can get the distance vector from our point to the light:
distance_vec = light_pos - point_pos
As well as the direction of the light for this particular point as a normal vector:
light_direction = distance_vec.normalize()
Then, using our knowledge about angle and dot product relationship, we can use dot product of point-to-light normal to calculate the brightness of the point. In simplest case it will be exactly equal the dot product!
brightness = surface_normal dot light_direction
Believe it or not, this is the bare bones of a simple light shader. An actual fragment shader in OpenGL would look like this (do not worry if you have no knowledge of the shaders: this is just an example to demonstrate the practical application of dot product):
Note that the way "dot" and "normalize" functions are used is the only difference from previous examples.
Example: Distance from a point to a plane
To calculate the shortest distance from some point to a plane, first get distance vector to any point of that plane, do not normalize it, and multiply it with plane's normal vector.
distance_to_a_plane = (point - plane_point) dot plane_normal;
Example: Is a point on a plane?
If it's distance to a plane is 0, yes.
Example: Is a vector parallel to a plane?
If this vector is perpendicular to plane's surface normal, then yes. We already know that two vectors are perpendicular when their dot product is 0.
So vector is parallel to the plane when vector dot plane_normal == 0
Example: Line intersection with a plane
Let's say our line starts at P1 and ends at P2. A point on plane surface SP and surface normal is SN.
If we make an imaginary plane run through the first line point P1, the solution boils down to calculating which point (P2 or SP) is both closer to P1 and more parallel to SN. This value for can be calculated by using a dot product.
dot1 = SN dot (SP - P1)
dot2 = SN dot (P2 - P1)
You can calculate "how much" it intersects the plane by comparing (dividing) these two values.
u = (SN dot (SP - P1)) / (SN dot (P2 - P1))
if u == 0, line is parallel the plane.
if u <= 1 and u > 0, line intersects the plane.
if u > 1, no intersection.
You can calculate the exact intersection point by multiplying line vector to u:
intersection point = (P2 - P1) * u
Concept: Cross Product
A cross product is also an operation on two vectors. The result is a third vector, which is perpendicular to the first two, and it's length is an average of the both lengths.
Note that for cross product, the order of arguments matters. If you switch order, the result will be a vector of the same length, but facing the opposite direction.
Example: Collision
Let's say an object moves into a wall at an angle. But wall is friction-less and object should slide along its surface instead of stopping. How to calculate its new position for that?
First of all, we have a vector which represents the distance the object should have moved if there were no wall. We are going to call it a "change". Then, we are going to assume object is touching the wall. And we will also need the normal vector of this surface.
We will use the cross product to get a new vector which is perpendicular to the change and the normal:
temp_vector = change cross plane_normal
Then, the final direction is perpendicular to this new vector and the same normal:
new_direction = temp_vector cross plane_normal
That's it:
new_direction = (change cross plane_normal) cross plane_normal
Now what?
With this article, I have tried to bridge the gap between the theory and the real practical application of it in game development. However, it means that I had to skip a lot of things in between.
But I hope the picture is a bit clearer. If anything, this walk-through may serve as a quick overview of the way Vector Math is used in Games.
The purpose of this article is to describe a way to simulate game physics in two dimensions. We will use an approach known as Verlet integration, go over the basics of moving points and building shapes to topics like collision detection and response. The reader should know the basics of vector maths, meaning addition, subtraction, multiplication with a scalar and the dot product. More than that is not required, however, for deeper understanding more knowledge of Euclidean geometry in two dimensions wouldn't hurt.
Verlet integration
First of all, what is Verlet integration? The Verlet integration is a way of numerically integrating the equations of motion. For this article, you really don't have to know what numerical integration means; basically, the Verlet integration describes the movement of a point trough time. There are different methods to do that - I guess, most of you know the Euler method:
If we merge these two equations into one, meaning we substitute the VelocityNew in the second equation by the right hand side of the first equation, we get:
There are different types of Verlet integration methods - Position Verlet, Velocity Verlet and Leapfrog. For this article, we will choose the position version, because it has some nice features that we're going to take advantage of.
The corresponding equation for position Verlet is actually not much different from the one shown above:
As we can see now, the term (PositionCurrent – PositionOld) in the Verlet equation replaces the velocity if we compare it with the Euler approach. Consequently, this means that this approach doesn't deal with velocities at all - one thing less to worry about. This integration method is not always quite accurate, since (PositionCurrent – PositionOld) is only an approximation of the actual velocity. However, it's fast and stable, which is why it is well suited for games. Also, using this approach the collision response gets really simple. Let us consider a point as shown in figure 1.
The starting conditions for PositionOld and PositionCurrent are chosen such that the point moves slowly to the right. After a certain amount of time steps, the point will intersect with the square on the right. Once the collision is detected, we only have to move the point out of the square and we are done with the collision response. Since the integration uses (PositionCurrent – PositionOld) as its velocity, the speed of the point will subsequently change if we change either CurrentPosition or OldPosition - which is what we did in the collision response (we moved the point out of the square). As we can see in figure 2, the point will automatically decelerate and eventually stop.
If we put the formula above in code, we get something like this (assuming you have a working vector-class, that is):
Now we have a working physics code that will calculate the trajectories of arbitrary points just fine. But points alone are not very useful, except when you're programming a particle simulation. Since that's normally not the case if you're into game programming, we have to extend the points in some way so we can simulate rigid body behaviour as well. If we look at rigid bodies in nature, we see that they are actually a huge amount of points (=atoms) held together by various forces.
We could of course try and create thousands of particles and connect them in some way to approximate the behaviour of rigid bodies, which would work indeed. However, that would cause a huge amount of calculations to be done for e.g. a single cube, let alone a whole game filled with physics bodies, so this isn't really an optimal solution. Luckily, it shows to be sufficient only to model the vertices of a body. If we were to simulate a box, we would simply create the four vertices that make up the shape of a box, connect them somehow and we're done.
The problem left to compute is now only that of the connections. If we again imagine a box and the four vertices, it should become clear that the distance of a vertex to another should always remain constant. If the distance between two vertices changes, this always means that the shape of the body gets deformed, and we don't want that - who would like to have a crate in his game that collapses once you stand on it? Therefore, we have to find a way to keep the distance between two vertices at a constant value.
If we had the same problem in reality, the solution would be simple - just insert some kind of pole in-between and the vertices won't approach each other anymore. We will do the exact same thing in our program; create a new class that represents an 'imaginary pole'. It connects two vertices and keeps those vertices at a constant distance. The algorithm to update these 'poles' is called once the Verlet is calculated. The algorithm itself is actually quite simple. First of all, we have to calculate the vector between the two vertices that are connected by the pole. The current distance between the two vertices is simply the length of that vector. Once we have the current length, the difference of the original length of the pole should be calculated. We can now use the difference to push the vertices to a position where the distance constraint is satisfied.
That's it - if we created a few points and connected them with our newly created edge struct, the resulting body would show very nice rigid body-behaviour, including rotational effects when it hits the floor. But why does that work? The code isn't much different from before, we only added a few lines to satisfy the distance constraint and suddenly we have rigid bodies. The reason behind this lies within our integration method. If we recall that the Verlet integration doesn't work with velocity but rather with the difference between the current position and the position before the last integration step, it should become clear that the speed of the point will change if we change its position. Therefore, since we change its position in the UpdateEdges method, its velocity will also change. The overall change in velocity looks exactly like we would expect it from a vertex of a rigid body; it is not totally correct, but good enough for games.
To be honest, I lied when I said before that the code would work just fine if we executed it like that. As the code is now, bodies would not be totally rigid. If a body collides with the floor, the distance between it's vertices is not totally constant, which means that the body is more or less deformed, depending on the it's speed before the collision. Why does that happen? The UpdateEdges method is totally correct, but still the distance between two vertices may vary. If we look at figure 3, this should become clear: If a vertex is connected to more than just one edge (which is normally the case), the length correction of one edge may disturb the length of another edge, which is why the bodies get deformed.
The only way to get rid of this problem is to execute the edge correction method more than just once per frame. The more this method is called, the more perfect the situation gets approximated, where all vertices have the right distance to each other. This gives game programmers a scalable physics algorithm - the more time is left at the end of the main loop, the more iterations can be used for the distance correction (and the collision response that will be introduced later). Vice-versa, if the main loop takes more time to execute, the iterations used for physics can be reduced so the game runs at a more or less constant frame rate.
Collision Detection
Now that our algorithm supports the simulation of (almost) rigid bodies, let's proceed to the next problem - collision detection! In this article, we will use an algorithm known as the 'Separating Axis Theorem'. If you already now how it works, you might as well just skip this part and go straight to the collision response. So, how does the Separating Axis Theorem work? As the title suggests, it states that two bodies don't collide, as long we are able to put a straight line between the two, that doesn't intersect either body. Figure 4 demonstrates this.
The only limitation of this algorithm is that it only works correctly with convex shapes. If we tested two concave shapes, the algorithm will fail, meaning that it would detect a collision when there is none. The reason should become evident if you take a look at figure 5.
We could deal with that by breaking up each concave polygon into convex subshapes and then test each subshape separately, but for simplicity reasons, we will just stick to convex polygons in this article - feel free to add concave support later.
So, how do we find out whether we could put a line in-between? We could of course just test every possible line for intersection, but it is evident that this is completely inefficient. To do this, we will take advantage of projection. If we put a new line in figure 4 that is perpendicular to the separating line, we can see that the projections of the two bodies on this line do not overlap (as shown in figure 6). However, if we chose a line that does intersect, the projections of the two bodies do also overlap.
It is irrelevant where we place the line that we project to, since the resulting projection is one-dimensional anyway - only its direction is important. This means that we don't have to look for a line that fits perfectly in-between the two bodies anymore, but for a direction where the projections don't overlap. Finding this direction shows to be quite easy. Let's consider the case where there is only one possible line that separates the two bodies (see figure 7).
It is evident that this line is parallel to the left edge of the right body. Therefore, to find the direction of a separating line, we simply have to iterate over all edges of both bodies and check, if the projection of the bodies onto the perpendicular of the edge overlap. If they don't, the bodies don't collide and we can end the search. If they do, we go on to the next edge. If there is no such edge, the entities are colliding and we have to proceed with the collision response.
Let's put this into code! But before we can implement the collision detection, we first have to write a body class that contains its respective vertices and edges:
The ProjectToAxis method will project the body onto the passed axis and change the Min and Max variables to the result of the projection. Since a projection of a 2D-shape onto 1D results in a mere interval of a line, the result of the projection can be stored in two floats that denote the beginning and the end of the interval. The projection method is quite simple:
The algorithm works just like described above; if something isn't clear, I would suggest rereading the explanations step by step. The IntervalDistance method that is mentioned in the code is actually quite simple:
Since we don't know if body A will lie on the left and body B on the right or vice-versa, we have to check which interval begins sooner. We then subtract the end of the left interval from the beginning of the right interval to get the distance between the two - if this value is smaller than zero, they overlap.
That's it for collision detection! ...well, not yet. Apart from detecting whether or not the two bodies collide, the collision detection should also provide certain information about the collision. We also have to calculate a so-called collision vector that is big enough to push the two bodies apart so they don't collide anymore, but touch each other. There are of course arbitrarily much vectors that could accomplish this, but for our physics to look right we have to find the smallest of those vectors. The vector we're looking for has the pleasant property that it's always parallel to one of the lines we projected to, which means that we only have to check each edge and calculate the length of the vector needed to push the two bodies apart. Figuring out the length isn't really a hard thing to do, if we take a look at figure 8.
In the code above we projected both bodies onto the axis given by the (normalized!) vector 'Axis'. We then called the method IntervalDistance to check whether or not the intervals are overlapping. The length of the vector (which is parallel to the axis we projected to) needed to push the two bodies apart is simply the amount of overlapping. To allow the information calculated in the DetectCollision method to pass smoothly to the collision response, we add a new struct to our Physics class:
Once we have this, we'd be already able to write a very simple collision response. Since the collision vector we calculated pushes the two bodies apart so they don't collide anymore, we could just move all vertices of both bodies back by half the vector and we'd be done. This would work, since interpenetrations are resolved, but it wouldn't look right. The bodies would simply glide off each other, meaning they don't start to spin like a real object when hit.
The problem is that a body in our approach only spins if the velocities of its vertices differ. In the same manner, a body only changes its rotational velocity if its vertices experience different acceleration. Acceleration is change in velocity, and in Verlet integration change in velocity is equal to change in position. Therefore, if we move the two bodies back by the collision vector, we change the velocity of all vertices of both bodies by the same amount, which means that there is no change in the rotational velocity. For this reason, we need to write a better collision response.
This is where the advantage of our approach kicks in! In a rigid body system, we would have to use complicated formulas to calculate the momentum and then treat the linear and angular case separately. In our system, the whole thing is much easier - we just have to move the edge and the vertex participating in the collision backwards so they don't intersect, but touch each other. Since both the edge and the vertex are connected to the rest of their respective body, the position (and therefore the velocity) of the other vertices will change immediately to fulfil the length constraint. Both bodies will start spinning self-actingly. The whole collision response reduces to identifying the edge and the vertex that participate in the collision and separating them from each other; everything else will be done automatically by the edge correction step.
Identifying the collision edge and vertex is not that hard. The collision vertex is the vertex that lies closest to the other body. Therefore, we simply have to create a line whose normal vector is the collision normal (its starting point doesn't really matter). We then measure the distance of each vertex of the first body from the line using the line equation in vector geometry, which is
where N is the normal vector, R0 the origin of the line, R the point to be tested and d the distance of the point from the line. The set of all points that form a line are given by d = 0 (all points that have zero distance from the line), just as a side note. Once we have the distance of each vertex from the line, we choose the one with the lowest distance - that's the collision vertex we were looking for. Please note that d can also be negative. A line separates a two dimensional space in two halves; if the point R lies in the half the line normal points into, the distance will be positive, but if it lies on the other side, the distance will be negative. Therefore, it is important in which direction the collision normal points (in the implementation presented below, it's always made sure that the collision normal points at the body containing the collision vertex).
The collision edge is even easier to find. Remember when we projected the bodies on the perpendicular of an edge to find the smallest collision vector? The collision edge is simply the edge that resulted in the smallest vector.
Time to put this in code! First of all, we have to extend the collision info struct in the physics class to contain the collision edge and vertex:
In the above code, we introduced a new variable in the PhysicsBody struct, the center. It will be recalculated before the collision step and is simply the average of all vertices of the body.
Collision Response
Finally, we're done with collision detection. The only thing left to do is the collision response, which is luckily not that hard. As explained above, we just have to push the collision vertex and the collision edge apart by the collision vector and we're done. This is trivial for the collision vertex. Since we already ensured that the collision normal points at the first body which contains the collision vertex, we just have to add half of the collision vector to the position of the vertex:
For the edge case, this will become a bit more complicated. The edge consists of two vertices that will move differently, depending on where the collision vertex lies. The closer it lies to the one end of the edge, the more this end will move and vice-versa. This means that we first have to calculate where on the edge the collision vertex lies. This is done using the following equation:
Where V is the position of the collision vertex and E1 and E2 are the two vertices connected by the edge. t is the factor that determines where on the edge the vertex lies, reaching from 0 to 1. It doesn't matter whether we choose the X or the Y coordinate to calculate t, since both would result in the same value. The X case would look like this:
This basically divides by the X denominator if it is bigger than the Y denominator and vice-versa.
We then use the following neat formula to calculate a scaling factor that ensures that the collision vertex lies on the collision edge after the collision response. We could derive it by solving a few equations, but I don't think the derivation is really important, so I'll just leave this here:
IterateCollisions is a method that does multiple things. It iterates over all bodies, calls the respective UpdateEdges method, recalculates the body center and then does the collision detection (and the collision response, if necessary). Of course, it doesn't just do this once, but repeats those steps a few times. The more repetitions are made, the more realistic the physics will look. The reason was explained above (if you've forgotten, better read it again ;) ).
Final Words
You can download a working implementation using GLUT and OGL as a Visual C++ 2008 project via the attached resource file. It is basically the same code as discussed in this article with a few optimizations and a very simple rendering and input function. I hope you enjoyed this article and found it useful. If you have any questions or suggestions, please let me know. Since I'm not a native English speaker, there might be a few mistakes here and there; please bear with it.]]>Thu, 19 Nov 2009 22:49:22 +00009e69fd6d1c5d1cef75ffbe159c1f322e2D Car Physics
Introduction
Someone asked me in the #gamedev IRC channel about how to make a 2d vehicle simulator. Instead of spending all day trying to explain the concepts to him, I decided just to write this tutorial.
Please bear with me, this is my first tutorial.
So as I mentioned we're going to be learning how to make a basic 2d vehicle simulator. We're going to do it in C# and try do use as few hacks as possible. I've broken the process
down into three steps. First, we will learn how to setup a basic game application in C#.NET and how to draw some basic graphics (emphasis on basic.) Next, we will learn how to create a rigid body
simulator using a simple Euler integrator with a variable time step. And last but not least, we will calculate vehicle forces simulating the tire patch contacting the road. And that's all there
is to it! Let's get started.
Math Requirements
There are two ways to get through this tutorial: you can rush to the end and download the project, or you can read through it and hopefully I'll be able to explain things clearly. If you choose
the second route, you're going to need to have a bit of math background. In a 2D simulation this is mostly in the form of a vector object. You'll need to be able to add, subtract, dot,
and project 2 vectors. Also you'll need to be able to use a cross product. In 2D this is kind of a fake situation since we know the result will point in the screen's direction, so the
result is returned as a scalar. If you're not familiar with any of these terms please look them up now. I tried to write this tutorial without using a matrix object but eventually I cracked and
used the Drawing2D.Matrix object to transform and inversely transform a vector between spaces. If you don't know what I'm talking about let me give you an example. Let's say your
personal body is your "local space" and the room you're sitting in is the "world space." Let's also say that your monitor is the front of the world, and your eyes
look in the forward direction of your local space. If you turn sideways, and transform the monitor's direction into your space, it is now the side direction. Vise versa, if you transform your
facing direction into world space, it is the opposite side direction. This is a critical concept so please, if that didn't make sense, do some searching on Google for transforming between
spaces. The reason this is so important is because we will be doing all of our vehicle force calculations in local vehicle space. Yet the vehicle itself, and its integrator, persist in world space.
Phase One
Phase one, as I mentioned, is to create the renderer; something graphical so we can actually see what our simulation is doing. This will make it a lot easier to debug. Create a windows form
project in C# and place a picturebox control on it (name it "screen"). This control is where we will display our simulation. We could just start drawing to this screen but we're going to be
using double buffering as well to avoid flicker, so we need to create the back buffer now. That bit of code looks like this.
The Init function must be called with the size of the "screen" control that you created on the form. This will create a bitmap "backbuffer" to which we can do our
offscreen rendering. We'll then take this backbuffer and draw it to the screen to illiminate any flickering. This is how you draw a basic shape to the backbuffer, and present it to the screen.
This function is called from the on_paint method of the "screen" control placed on our form. The on_paint method has a parameter "e" that contains a graphics
object we can use to draw to the control. We pass this graphics object to the render function and as you can see, we draw the backbuffer to it as the very last step.
Now by default, the graphics of a picturebox control has the origin in the top-left corner, and extends downward for +y and to the right for +x. This is highly unnatural for most cases. In
addition to that, it has extremely large units. Since we will be simulating in the metric system, I recommend introducing a scale factor to scale up the simulation and make it much more visible. The
transformation looks like this and takes place after Graphics.Clear() is called.
That transformation flips the Y axis so that +Y points up. It simultaneously scales the space by our "screenScale" factor (something like 3.0f should work fine). Next, we translate the
graphics space into the center of the screen control by half of the screen dimensions divided by our scale (since we are now in the scaled space.)
Now the line should draw starting right at the center of the screen.
Forms Wiring
Up until now, I havn't explained how to connect all the functions. The first thing you'll need to do is call the Render function from your on_paint event.
Next, you'll need to create a function that gets called continously to update the simulation. It is preferred to call this function on the Application_Idle event. So create an
event handler for Application_Idle and have it call your DoFrame function. Inside this function you'll need to
Process input
Update the simulation
Invalidate the screen control
The last step is so that an On_Paint gets triggerd and the simulation gets drawn. You'll also want to wire up some "key_down" and "key_up" events to keep
track of key states.
The Timer
Since we don't know how often our DoFrame function will be getting called, we need to code everything to handle a variable time step. To utilize this we must measure the time
between DoFrame calls. So I'll introduce the timer which, very simply, queries the number of milliseconds that have passed since the computer was turned on. So we store this number
every frame and on a subsequent frame we compute the difference, which gives us the amount of time that has passed since the last frame. Here is my very simple timer object. Note: you will
need to call GetETime in your intialize function in order to clear the timer, otherwise the first call to it will return the amount of time that has passed since the computer was turned
on.
Conclusion of Phase One
So up until now we've covered: setting up a rendering surface using GDI, wiring a form to process a game loop and draw it to the screen, and computing the time that has passed since the last
frame. Our application looks like this:
Phase Two - Rigid Body Simulation
Ok, now we're getting into some good stuff here. Let's put everything we just covered on the back burner now and talk about some physics. We're going to be using a very simple Euler
integration method. Basically, each frame we accumulate a bunch of forces (in our case from each wheel of the vehicle) and calculate the resultant acceleration, which is in the form of A=F/M (the
same as F=MA, Newton's second law of motion). We use this to modify Newton's first law of motion, "an object in motion stays in motion…" So we calculate our A, and we
integrate it into our V. Without an A, V would be constant, hence staying in motion, if no forces should act on it. Newton's third law gets applied in the form that any potential force the vehicle is
applying to the ground, gets applied in the opposite direction to the vehicle (I'll explain this in the vehicle section). This topic is much easier to explain with symbols. So, P is our vehicle
position, V is its linear velocity, F is the net force acting on it, M is its mass, A is the resultant acceleration, and T is the time step (the value our timer gave us from the last frame).
A = F / M
V = V + A * T
P = P + V * T
So with a constant mass, and some force, we will generate acceleration, which will in turn generate velocity, which will in turn generate a displacement (a change in P).
This is a basic linear rigid body simulator. Each frame, we total up some F, integrate it, and then zero out F to restart the accumulation the next frame. Now let's talk about rotation. The
angular case is nearly identical to the linear case (especially in 2D). Instead of P we have an Angle, instead of V we have an Angular Velocity, instead of F we have a torque, and instead of M we
have inertia. So the angular model looks like this
Simple huh? Now you may be wondering where this Torque came from. A torque is generated every time you apply a force. Lay a book down on your desk and push on the corner of it. The book should slide
across the desk, but it should also begin to rotate. The slide is caused by the force. This rotation is caused by the torque, and the magnitude of the torque is directly proportional to how far away
from the center of the object the force was applied. If you applied the force directly to the center of the object, the torque would be zero. We need to construct an AddForce function
for our rigid body. This is what gets called every frame, once per wheel, to accumulate the chassis' rigid body force/torque. The linear case is simple, Force = Force + newForce. The angular case is
a little trickier. We take the cross product of the force direction and the torque arm (the offset between where the force was applied and the center of mass of the body.) In 2D, this results in a
scalar value that we can just add to Torque. So, Torque = Torque + TorqueArm.Cross(Force)
This is what that bit of code looks like. % is the cross product operator for my vector class.
You'll notice the "world" prefix on the parameters. This is because all computation of the rigid body happens in world space. So as your book is rotating on the desk, the worldOffset
value is changing, even though your finger is not moving on the book (this would be the relativeOffset). So if we know we're applying a force "across the book, at the top right corner" we
need to convert both "across" and "top right corner" into world space vectors, then add them to the rigid body.
Code Dump
Here is my rigid body object. You'll notice all the properties I mentioned above. It has a Draw function which will draw its rectangle to the provided graphics object. It has an AddForce function, a space conversion method, to and from world space (very handy), and a function that returns the velocity of a point on the body (in world space). This point velocity
is a combination of the linear velocity and the angular velocity. But the angular velocity is multiplied by the distance the point is from the center of rotation and perpendicular to its offset
direction. So to kill two birds with one stone, I simply find the orthogonal vector to the point offset and multiply it by the angular velocity (then add the linear velocity.)
One thing you may be curious about is how I calculate the inertia value. That is a generalized formula I found at this
link.
Testing
To make sure your rigid body works, instantiate one in your Init() function and apply a force with some offset in the DoFrame function. If you apply a constant
worldOffset, the body will continue to accelerate its angular velocity. If you take your offset and run it through the RelativeToWorld function, the body will angularly accelerate in one
direction and then come back the other way, like a pendulum as the point the force is applied to changes. Play around with this for a while, this has to work and make sense in order for the next
section to work.
Phase Three - The Vehicle
Assuming everything has gone well above, you should have a rigid body actor in your scene that you can apply forces to and watch move around. Now all that's left is to calculate these forces
in a way that will simulate a vehicle. For that we are going to need a vehicle object. I recommend deriving directly from you rigid body object since the chassis is essentially a rigid body. In
addition to that we will need to construct a "wheel" object. This wheel will handle the steering direction of each wheel, the velocity the wheel is spinning, and calculate the forces that that
particular wheel applies to the chassis (all in vehicle space). Since our wheel is known to be constrained to the vehicle, we don't need to simulate it as another rigid body (though you could,
but not in the 2D case.) We will simply duplicate the angular properties of the rigid body in the wheel object.
So we'll need: Wheel Velocity, Wheel Inertia, and and Wheel Torque. We'll also need the relative offset of the wheel in the vehicle space, and the angle the wheel is facing (this is
constant for the back wheels, unless you want 4 wheel steering.) Just like the rigid body, the wheel's torque function acts as an accumulator, we add torques to it and after it gets integrated the
torque is zeroed out. The AddTorque function is where you will apply a wheel torque from either the transmission (to make you go) or from the brakes (to make you stop). Internally the
wheel will generate a torque caused by the friction on the road.
The wheel object also needs a SetSteering function. This function calculates two vectors: an effective Side Direction, and an effective Forward Direction (both in vehicle space) that
the tire patch will act on. The force applied on the tire by the ground acting in the side direction will directly translate into the chassis. Meanwhile the force acting in the forward direction will
not only act on the chassis, but it will induce a rotation of the tire. Here is the SetSteering function; you will see I used the Drawing2D.Matrix to transform the initial
forward and side vectors by the steering angle (I had to convert the vectors to "points" in order to transform them by the matrix.)
Force Calculation
So, if the vehicle is sitting there not moving with its front wheels turned, and you push it, a force will be generated in the opposite direction you push. This force gets projected onto these two
directions. If the wheels were straight there would be no side force. So the vehicle would simply roll forward. But since the wheels are turned, there is a bit of the force that acts in the
"effective side direction" so we apply an opposite force to the chassis. This is what causes you to turn when you steer the wheels. To get this force that gets projected onto the two
directions, we need to first determine the velocity difference between the tire patch and the road. If the wheel is spinning at the same speed the ground is wizzing by, then there is effectively no
force acting on the vehicle. But as soon as you slam on the brakes and stop the wheel, there is a huge velocity difference and this is what causes the force that stops your car.
Almost Done!
We're in the home stretch here now. Now we have a way to calculate the force each wheel generates on the chassis. Every frame, all we have to do is set our transmission and brake torques,
our steering angle, calculate each wheel force, add these to the chassis, and integrate the rigid body. Badaboom badabing, vehicle done! :)
Conclusion
Here is the entire source code for the project. If you have any questions or comments please
feel free to post them here and either I can make things more clear or maybe someone else could offer some better expertise. If you'd like you can email me at Kincaid05 on google's fine
emailing service.
Thanks for reading and I hope this was informative.
-Matt Kincaid
]]>Sat, 15 Dec 2007 02:36:46 +00008e9cd191d3eaf58c4d262677292270e5Randomness without Replacement
"Game mechanics" is one of those terms that every designer talks about. Everybody agrees that game mechanics is an important subject within game design. Yet almost nobody discusses, in detail, how the design of a mechanism may satisfy a player. In this article, I'm going to dissect a mechanism germane to role-playing games, the randomization function for determining whether a player's attack hits or misses its target. Roll up your sleeves for a modest amount of mathematics—just enough to demonstrate what a game mechanism is.
Reducing Frustration In the opening story, one way to prevent frustrating results is to reduce randomness. Several massively multiplayer role-playing games, such as Lineage 2, reduce variance of hits and misses and reduce the variance of damage. Reason? As Alex Chacha stated, if the probability for a string of failures is above zero, then the probability for that string occurring at least once in the lifespan of a long game is very high ("Randomness," MUD-Dev Mailing List, February 2004.). And once it does, it is a frustrating experience. So, in this article, let us call such a long string of consecutive failures frustration.
To reduce the occurrence of frustration, one possibility is to change the mechanism that models the randomness. For instance, let's change the method from sampling with replacement to sampling without replacement.
But first, to understand this lingo, we need to know a little probability theory. It shouldn't be too hard for us, since probability theory was commissioned, in 1654, as the science of dice and card games. So we're in familiar territory: the mathematics of dice and cards. In probability, when a result of a random function is taken, it is called sampling. There are two basic methods for sampling: with replacement (as in a die roll), or without replacement (as in drawing cards from a deck). The difference seems simple, but the implications, as we shall see, may be dramatic.
Back to the problem at hand: To reduce frustration, change from sampling with replacement to sampling without replacement. Basically, instead of rolling a die, draw a card. Let's use Open Game Content (OGC) as shared vocabulary. This is the system that Wizards of the Coasts based Dungeons & Dragons 3.5 on. So, instead of rolling a twenty-sided die (1d20), draw from a deck of twenty cards, perfectly shuffled.
The key difference in the mechanism may be illustrated by a deck of cards alone. With replacement, after each draw, the card is shuffled back into the deck. Without replacement, the drawn cards are then discarded. Only after the deck has been consumed is the discard pile shuffled to reconstitute a full deck. To keep the analysis simple and further reduce frustration, remove one of the hit cards from the deck, and shuffle the remaining 19 cards. Then place this hit card on top of the shuffled deck.
Suppose, in OGC terms, that the player has a melee attack of +0, then he would hit armor class 11 about 50% of the time. This may be modeled, with replacement, as a probability of success of 50%. To model this probability, without replacement, imagine there are 10 hit cards and 10 miss cards. If details are required to account for modifiers, then there may be 20 cards, valued from 1 to 20. Either way, 10 of these cards will result in a hit, and 10 will result in a miss. Without replacement, the player could sample 10 misses in a row, but the probability of this frustration is much less than by sampling with replacement. But how much less probable is it?
Counting Cards Calculating the probability of frustration with replacement is simple. Suppose f equals the number of consecutive misses that will result in frustration, which we shall consider at 10 consecutive misses. With replacement, this is equivalent to a binomial distribution with a given probability of missing being the measure of consecutive trials. With a probability of a single failure being 50%, we may understand Equation 1. This logic is simple. The probability of failure ten times a row is the multiplication of the probability of failure ten times. So, in a sequence of ten attacks, the probability of frustration is about one in a thousand.
Equation 1. The probability of frustration, with replacement.
Calculating the probability of frustration without replacement is complicated, because the probability of each attack hitting depends on the success or failure of the previous attacks. A general method to solve problems of discrete probability is to count all the ways in which the event may occur, and divide this number by the count of all ways in which any outcome may occur.
In order to count possible occurrences of frustration, recall that the probability equals the number of ways to have a consecutive string of 10 misses out of 20 attacks. To simplify analysis and further limit frustration, set the first attack to be one of the hits. This corresponds to rolling a natural 20 in OGC. Since frustration (i.e., the string of 10 misses) is immutable, this is equivalent to the number of ways to select a single string of 1 out of 10. Recalling the basics of combinatorics (which is the mathematics of counting), this simplification makes computation trivial, as shown in Equation 2.
Equation 2. How many ways an attacker may be frustrated, without replacement.
Then count the total number of ways to have 9 hits and 10 misses in a sequence of 19 attacks, as shown in Equation 3.
Equation 3. How many ways an attacker may attack, without replacement.
By dividing the count of frustrations by the count of all attacks, we arrive at the probability of frustration, as demonstrated in Equation 4. So, the probability of frustration in the course of twenty attacks is about one in ten thousand.
Equation 4. The probability of frustration, without replacement.
By dividing the solution of Equation 4 by Equation 1, we see that the sample without replacement decreases the probability of frustration by an order of magnitude, shown in Equation 5. To keep the analysis simple, multiply the probability of frustration with replacement by 2 (or divide without replacement by 2), since the probability in a sequence of 20 attacks (without replacement) is being compared to a sequence of 10 attacks (with replacement).
Equation 5. Nonreplacement dramatically decreases frustration.
Clearly, this mechanism reduces frustration. Furthermore, the mechanism without replacement guarantees never to have more than 10 consecutive misses (since the 1+20kth occurrence always hits).
Although the above analysis is only a solution for one of the parameters that a player may have of hitting an opponent (i.e., melee attack of +0 versus armor class 11), the same analysis can be carried out on all possible parameters. Likewise, this analysis can be carried out for any given lower bound to frustration. If worst-case analysis of player satisfaction had predetermined 5 consecutive misses to be the maximum tolerance before frustration occurs, then the deck could be divided into two separate decks, each without replacement, so long as each deck had no more than 5 in it. For the sake of uniformity, two decks might be evenly divided (i.e., 75% success rate = 7 out of 10 and 8 out of 10).
God Does Not Play Dice—He Plays Cards This deck mechanism is not without its costs. A little detour from the mathematics department and into the computer science department will explain why. Each mechanism is implemented from an algorithm (which is a precise procedure for the rules of the game). Computing the amount of time required for the die mechanism is straightforward, as listed in Table 1.
Step in the algorithmComputations For 20 attacks: 20 Roll a die. dTotal Time Complexity 20 d Table 1. Algorithm outline of a die mechanism. The deck mechanism is not much more complicated. A naive shuffling algorithm can shuffle in linear time with one random call per element. So, this implies a running time not worse than 19 fold of the running time of the die mechanism. This shuffling only needs to be called once every 20 attacks. As listed in Table 2, the time complexity (which is an abstraction of the amount of time required to execute the algorithm) for a card mechanism is greater than the die mechanism, but not by much. Although there are two operations to perform instead of one, the drawing of a card (c) from a shuffled deck is a trivial operation. In addition, within an efficient shuffling algorithm, shuffling one card in a deck of 19 (s) is approximately comparable to rolling a die with 19 sides (d).
Step in the algorithmComputations Shuffle a deck of 19 cards. 19 s For 19 attacks: 19 Draw a card. cTotal Time Complexity 19 c + 19 s = 19 (c + s) Table 2. Algorithm outline of a card mechanism. A more elegant algorithm may shuffle one card per attack, which means one randomization per attack, as listed in Table 3. This does not alter the total time complexity, but does distribute the computations more evenly, so that there is not a delay (for shuffling) after each 20th attack.
Step in the algorithmComputations For 19 attacks: 19 Draw a card. c Shuffle that card. sTotal Time Complexity 19 (c + s) Table 3. Algorithm outline of a gradually shuffled card mechanism. Of course, compared to the die, the deck requires memory, but this may be as little as an additional 19 bits per player to as much as 50 bytes per player.
Albert Einstein once remarked on another detailed mechanism, a mechanism for the elementary fabric of the universe. When presented with the random mechanism for modeling quantum physics, Einstein said, "God does not play dice." In the case of our RPG, maybe the Creator plays cards, instead.
Does it Matter? Does it matter whether a die or deck is used? Both probabilities are negligible, making it unlikely for a designer to encounter such frustration. Yet even small probabilities, such as one in a thousand, are worth considering given enough players.
To understand, first we need to learn another distribution common to discrete probability. We want to know how many instances of player frustration will occur. Each instance is rare, so the mean of a Poisson distribution, E(X), is a good approximation, as formulated in Equation 6.
Equation 6. Poisson distribution models the count of frustrations.
In general, the Poisson distribution is used to measure the number of occurrences (λ) per unit time. We have already calculated the probability of a single instance of frustration (p). What remains to be determined are the number of attack strings in which frustration may occur (n). Let's consider two scenarios with sufficient sequences of player attacks, both in massively multiplayer games and in single-player games.
Case Study: Massively Multiplayer RPG For a simple example in a massively multiplayer role-playing game (MMORPG), there could average 1000 attacks per player per hour. To be crude, this could be coarsened to 100 strings of 10 attacks (Actually, only one of the ten possible combinations of ten consecutive failures conform to this demarcation). If—admittedly oversimplified—there were, for a reasonably small MMORPG, an average of 100 simultaneous players per day with uniform distribution of attacks, and independence of attacks, then there would be 100 strings / hour (24 hours / day) 30 days / month. The arithmetic equals 72,000 strings per month. Since this is very large and the probability is very small, the Poisson distribution approximates, with a rate of λ = n p. The expected number of frustrations is solved in Equation 7.
Equation 7. Expected number of frustrations per month, with replacement.
So there's going to be, on average for a low-traffic MMP, given these simplistic preconditions, 70 strings of frustration (i.e., 10 consecutive misses) per month using a mechanism analogous to a 1d20 die roll. Whereas, without replacement (a mechanism analogous to a 20-card deck), the expected number of frustrations is computed in Equation 8. For approximation, the number of trials is divided by two, because this is a sequence of 20, instead of a sequence of 10.
Equation 8. Expected number of frustrations per month, without replacement.
So, there are about 18 times fewer occurrences of frustration. A distribution of the frustration may be estimated by the Poisson distribution, as shown in Figure 1. Because only whole numbers can occur, the distribution has been stepped at the rounded (midpoint) value of a continuous Poisson distribution.
Figure 1. Distribution of frustration without replacement (blue) or with replacement (magenta).
From comparing the distributions, it is obvious that without replacement, there are almost certainly going to be fewer cases of frustration.
Case Study: Single-Player RPG There's no theoretical or practical reason why such methods could not be applied to single-player and multiplayer RPG combat. The fact is, a game that sells a million copies, can expect similar figures to an MMP. It is a difference of penetration and time. The same Poisson distribution applies. Suppose the average player of a single-player RPG spends 10 hours with the game (since many more players quit sooner than those that play longer). If the rate of attacks remains constant (1000 attacks per hour), then only 7200 copies of the game need be sold to have the equivalent of a month of the modestly populated MMORPG above.
Every Mechanism Should be Designed as Simple as Possible, but Not Simpler Einstein also said, "Everything should be made as simple as possible, but not simpler." Don't let mathematical precision dominate your analysis of a mechanism. While understanding the subtleties of a mechanism can make or break the design of a game, the relevance of the number crunching must be maintained.
In this example, we computed the difference, not in terms of all kinds of player frustration that may occur, but only with one single variation to a mechanism within the game. We simplified the situation in order to keep the math brief. We did not consider players with probabilities to hit at above or below 50%, nor did we strictly adhere to the definition of independence for a Poisson distribution in the die mechanism. Astute readers will have also noted that the deck mechanism does not randomize the first card drawn. Doing so would further complicate the calculation of probability. Therefore, our result is only an approximation for a special deck.
A more precise analysis was not necessary to prove the fitness of the deck mechanism. Bear in mind that a designer can correctly solve a problem but fail to solve the right problem. In this article, we honed in on one mechanism, ignoring the rest of the game and its effect on the player's satisfaction. Even if it were mathematically tractable, computing the distribution of loss of players due to this careful definition of one type of frustration is a tougher problem.
About the Author David Kennerly directed five massively multiplayer games in the US and Korea. He localized Korea's first world, The Kingdom of the Winds, and designed the social system of Dark Ages: Online Roleplaying. Before joining Nexon in 1997, he designed The X-Files Trivia Game for 20th Century Fox, and troubleshot US Army networks in Korea.
David encourages creativity among developers and players. He helped organize MUD-Dev Conferences, and founded an online library of fan fiction. David has authored on game design for Charles River Media, ITT Tech, Westwood College, Gamasutra.com, and IGDA. To discuss this article with the author, please visit his website
]]>Wed, 16 Feb 2005 16:11:45 +0000dfcebbaf79842c2e6fca7b77741de3a6A Simple Time-Corrected Verlet Integration Method
Introduction Verlet integration is a nifty method for numerically integrating the equations of motion (typically linear, though you can use the same idea for rotational). It is a finite difference method that's popular with the Molecular Dynamics people. Actually, it comes in three flavors: the basic Position, the Leapfrog and the Velocity versions. We will be discussing the Position Verlet algorithm in this paper. It has the benefit of being quite stable, especially in the face of enforced boundary conditions (there is no explicit velocity term with which the position term can get out of sync). It is also very fast to compute (almost as fast as Euler integration), and under the right conditions it is 4th order accurate (by comparison, the Euler method is only 1st order accurate, and the second order Runge-Kutta method is only 2nd order accurate [go figure]).
The disadvantages of the Verlet method are that it handles changing time steps badly, it is not a self-starter (it requires 2 steps to get going, so initial conditions are crucial), and it is unclear from the formulation how it handles changing accelerations. In this paper we will discuss all of these shortcomings, and see how to minimize their impact. The modified Verlet integrator is referred to as the Time-Corrected Verlet (TCV) and is shown below with its original counterpart. The computations used to generate the graphs for this paper are included in this Excel file.
To see why the TCV is an improvement, we need to see the math behind the original Verlet method.
Math In this paper we will be talking exclusively about point masses, which are acted on by forces. Well, we all know about Newton's little equation: Force=d(Momentum)/dt. Momentum=mass*velocity and for non-relativistic speeds the mass is constant, removing our need for the chain rule, and yielding our familiar F=ma. So if all the forces acting on the point mass are summed (the vector F), then scaled by (1.0/m) we have the acceleration (a) of the point mass. Since we know how to go from force to acceleration, we will be starting from the acceleration term to keep the math less cluttered. In actual applications you will almost always be summing forces, then converting to accelerations.
Most of the graphs presented in this paper include the matching Euler simulation, just for reference. The Euler algorithm is extremely simple, and it will not be derived here. However, the equations are included below for your reference (order is important):
v = v + a * dt x = x + v * dt
There is a more accurate version, but it is not strictly the Euler method, so it will not be used for this paper. However it does give 2nd order accurate results, instead of merely 1st order:
x = x + v * dt + 0.5 * a * dt * dt v = v + a * dt
Please note that there is a faster way to derive the Verlet method's math* than what will be shown here, however it does not provide the insight needed to overcome the issues mentioned in the introduction. So we'll start from a few basic principles: a=dv/dt, and v=dx/dt. So for any given point mass, if we know the current position, velocity and acceleration (the acceleration must be constant over the given time step), we can compute exactly where it will be after the time step (dt) has elapsed.
(1) xi+1 = xi + vi * dti + 0.5 * ai * dti * dti
or
(1a) xi+1 - xi = vi * dti + 0.5 * ai * dti * dti
Now, if we wanted that equation formulated without the velocity term, we could replace it with some other known state variable, such as the position x. But since we don't know xi+1 yet, we shift the whole equation back a step:
For this next step we need to make a big assumption, the importance of which will be seen later: If we assume that neither the acceleration nor the time step vary between steps (i.e. that ai-1 = ai = a and that dti-1 = dti = dt) then we note that:
You'll notice that the right hand side of equation (5a) is exactly the last half of equation (1), so we can work the modified equation (5a) back into equation (1):
(6) xi+1 = xi + (xi - xi-1) + a * dt * dt
and there you have the traditional Verlet Position integration method.
Fundamental Problems As you saw from the derivation's step (5), the two criteria needed to make the Verlet algorithm exact are constant acceleration and constant time step. For most practical cases we cannot guarantee either of these criteria. Of course, there are some simple cases where both criteria will be met. For example, simple projectile physics simulated by a physics engine which uses fixed time steps will yield perfect results. As soon as you add friction, springs or constraints of any kind you nullify the constant acceleration criterion, and adapting your time step to your game's framerate will nullify the constant time step criterion.
I am not going to address the constant acceleration criterion, mainly because explicit integrators (such as this one) must assume the constant acceleration principle, which is violated the instant you start simulating a complex system. Take the example of a point mass connected to a spring: as soon as it starts to move, the spring force, and thus the acceleration, changes. Even equation (1) required that the acceleration be constant throughout a time step. The error introduced by assuming that ai-1 = ai is actually implicit in our choosing an explicit scheme without knowing how the acceleration changes. We are effectively setting d(a)/dt (a.k.a. the "jerk") to 0.0. The other reason I will ignore this issue is that, empirically, the standard Verlet method already handles changing accelerations better than the Euler method (or even the improved variation on the Euler method), as long as the time step is fixed. Note that the Time-Corrected Verlet will be identical to the original Verlet when the time step is fixed. Observe the following graphs:
The error introduced by the constant time step assumption is something that can be easily ameliorated. This will be shown in the section entitled "A Simple Time-Correction Scheme".
Implementation Problems A large source of inaccuracy when using the Verlet scheme stems from the improper specification of initial conditions. Looking at equation (6) and trying to fit it into the form of equation (1) (which may be more familiar) may yield (improper) reasoning like this: the first (xi) term is the position contribution, the second (xi - xi-1) term is basically the velocity contribution, and the (a * dt * dt) term is clearly the acceleration contribution. So when simulating the traditional projectile path, setting x0 = 0.0, and x-1=x0 - v0 * dt0, and running the simulation from there will give the wrong results, as can be seen in the following graph:
This is because both the second and third terms include acceleration information. So, when setting the current state explicitly (i.e. the position and velocity initial conditions), remember to use equation (2). Shooting simulations depend upon starting with accurate initial conditions. The larger the initial time step, the less accurate your computed initial state will be if you do not use equation (2).
A Simple Time-Correction Scheme The remaining fundamental problem with the Verlet integration method lies with its assumption of a constant time step. The (xi - xi-1) term from equation (4a) is the portion of the equation which is dependent on the constant acceleration and constant time step assumptions. It has been explained (OK, hand-waved away) why the usage of the last time step's acceleration will not be corrected, but we still have the problem of dti-1 being stale information. Rewriting equation (4) yet again, we see that:
(4b) xi - xi-1 = (vi - 0.5 * ai-1 * dti-1) * dti-1
so taking the easy way out and ignoring the dti-1 linked with the acceleration, I can swap out the old dti-1 for my new dti by multiplying (4b) by (dti / dti-1). Plugging all of this in yields my final form of the Time-Corrected Verlet integration method:
(7) xi+1 = xi + (xi - xi-1) * (dti / dti-1) + a * dti * dti
You will notice that when the last frame's time step equals the current frame's time step, the modifier term becomes 1.0, yielding the traditional form of the Verlet equation. As an optimization note, only one value needs to be stored per frame, as the dt was constant for all points in the last frame. So the term (dti / dti-1) can be computed once per frame and stored in a variable. At the end of the Verlet update subroutine simply store the current frame's dt as old_dt. Likewise, dti * dti can be computed once and stored in a variable, saving a multiplication.
To show how the Time-Corrected Verlet behaves, a spreadsheet was set up with the TCV, the original Verlet and Euler's method, each simulating three different problems with known solutions. These are the same tests as were performed earlier, but with randomized time steps. Of course since every time step (except the first) was random, the graph looked different each time a simulation was "run". Sometimes both the original Verlet and the TCV simulations were similar, however the original Verlet always fell further away from the exact solution than the TCV version eventually. Here are some sample graphs:
Conclusion Using the Time-Corrected form of the Verlet integration method with the proper equation for initializing the state makes the TCV integration scheme a simple, yet powerful method for doing game physics, even with changing frame rates.
It restores some of the accuracy of the method (still 4th order with constant time steps, between 2nd and 4th with changing dt), while maintaining its cheap numerical cost. I hope this helps a bit when implementing your own physics simulation code.
]]>Sun, 06 Feb 2005 23:50:36 +0000b62973ddb1a4d7a0ea30a1052012af19Opposing Face Geometry
Abstract: OFG presents a new method for collision detection optimizations by performing a simple pre-calculation on both input objects. It is possible to reduce the number of faces necessary to check for intersection dramatically, from an order of O(mn) intersection tests to an order of O(k2), or rather to a maximum of k2 + 3k test operations, where k is a predetermined constant. The pre-calculation phase is of the order of O(m + n). Therefore, for increasingly complex convex objects, the OFG method saves more and more processing time. The method's downside is that increasingly complex objects might need a very high constant and small faces are less suited for this type of optimization. The method is much better for relatively low detail 3D objects.
Introduction Collision detection is generalized as the means to detect whether any two objects in 3D space overlap. Over the years many models and ideas were suggested that attempt to either solve the problem entirely or approximate a solution. Of course, it has been shown that detecting collisions in a very accurate way is extremely computation intensive. Yet new ways and methods have been invented to optimize or accelerate the collision detection algorithms so those will be useable in real-time environments such as 3D simulations or 3D games. The method proposed here is called OFG and attempts to do the exact same thing - to optimize the collision detection algorithm by eliminating as many checks as possible.
The problem In real life, collision detection is a fact, it's simply how things work. Objects cannot occupy the same space, at least not usually. However, when dealing with computer programs it is clear that it's impossible to simulate the detail levels of the real world. In computer simulations the only way to actually define a 3D object is by defining the points from which its polygons are made. Each of these points is called a vertex, in plural - vertices. Defining a 3D object using only points connected by lines is the only way to represent a 3D object in a way that allows real-time simulations. This method is of course only an approximation, but when there are enough polygons making up an object, the approximation is very good.
The representation of a 3D object is very simple. Vertices can be connected by lines to create closed shapes called 'polygons'1(usually triangles). A collection of faces constitutes a closed 3D object and therefore the only information at our disposal for collision detection testing is the vertices information. The problem begins by trying to create a model for collision detection. It seems highly unlikely that objects collide at exactly their vertices, and this is logically correct as well. Consider two normal cubes the same size. The two cubes can collide in an infinite number of ways and orientations, even without their vertices touching each other's or any connecting line between vertices (edges)2. What is more logical is that either edges themselves pierce the other object's faces, or some variation to that effect. The question then arises, how to detect faces intersecting each other? or better yet, how to detect edges piercing other faces?
Well, the good news is that there are already good ways of testing for edges piercing faces. The bad news is that representing complex 3D objects requires a great deal of faces to look reasonably well, and since detecting intersection in an accurate way is slow in comparison to other optimized methods, this creates a big problem. In this article we will assume that detecting whether two faces intersect is a problem solved in a reasonable way and hence this article will not deal with methods for detecting actual intersection of a pair of faces, which is the most elementary test. For convenience a few methods are given in the appendix but it must be clear that OFG is a method for optimizing the entire process of object to object collision tests by dramatically reducing the number of faces needed for testing.
Basic assumptions and the most general case In order to simplify the problem some basic assumptions need to be made. Of course these might present some problems but it must be clear that the assumptions help solve the most simple case. Later on the algorithm will be improved to circumvent some of the problems arising from the assumptions.
The assumptions that we make are as follows:
The objects that the algorithm is dealing with are convex3 3D objects only. It is true that the algorithm will work for some concave4 objects but care must be taken5.
Detection of any pair of faces colliding is a well-defined and solved problem.
Both objects have a pre-calculated center point.
The accuracy constant k is set to be (min/max estimates)
Object A has exactly n faces while object B has m faces.
Under these assumptions, in order to detect a collision between objects, the most simple but inefficient method of checking every face in object A against every face in object B can be used. Actually, using the brute force method works for all types of objects, convex or concave. It makes no difference to the algorithm. For the brute force method therefore an order magnitude of O(nm) represents the algorithm's time complexity. The problem with this approach is obvious - detecting collision between complex objects becomes a very long operation. The more faces objects have, the more checks are needed. For two normal cubes, each with 6 sides (two faces per side, assuming each face is a triangle), the number of checks is: 12 * 12 = 144 tests between pairs of faces.
For two objects with 100 faces each, the number increases very fast: 100 * 100 = 10,000
For two objects with 200 faces each, the number increases again to: 200 * 200 = 40,000
It is clear that this method is highly inefficient when dealing with more and more complex objects.
The Opposing Face Geometry algorithm OFG is a method that at its basic level attempts to find in the simplest way the closest geometry elements two objects have. The algorithm attempts to find the closest faces both objects have in relation to one another while the number of desired faces to be found is determined by the accuracy constant k.
The OFG method consists of the following steps, each described in more detail in the following sections:
Calculate the geometric center of object B's new selection of faces and their maximal bounding sphere radius.
Optimization: check collision between spheres to determine if there is even a chance for face collisions.
Sort the two sets of faces by increasing distance (optional, might be replaces by a good insertion algorithm).
Test the two sets of faces against each other, starting with the closest pairs of faces.
Analysis:
It's clear from the above steps that there are some preliminary preparations before any actual checks are done between faces. The time complexity of steps 1, 3, 5 and 6 is rather fixed: 2 operations, 3k operations, 3k operations and 2 operations again respectively and therefore contribute only little overhead to the entire process. Step 7 is basically a sorting operation for an array of k elements. If care is taken to insert the elements in an almost-sorted fashion, it is possible to use sorting algorithms that operate at almost O(n) on each of the arrays, or better. Since n=k in this case, O(2k) can be added to the overhead of the algorithm however up until now the overhead presented by these steps is rather constant no matter the complexity of the objects and is said to be geometry independent.
It will be shown that step 2 requires a time complexity of O(n) while step 4 requires a time complexity of O(m). Therefore the geometry dependant time complexity (and hence the most important one) is O(n+m).
As for step 8, the worst case scenario of testing k faces against k faces is O(k2).
Note: The accuracy constant k plays a major role in the OFG algorithm. The higher k is, more faces are selected for testing and therefore more accurate collisions can be detected. Typical values are from 4 to 8 faces and the reasoning for this is explained later on in the appendix.
Step 1 - Checking for the possibility of a collision There is no real need to test any faces at all if the objects are far away from each other. This is logical but presents a problem: since objects are almost always non-spherical, one has to define a range from which no face testing will be performed and for a shorter range, face testing will be performed. But if the objects are non-spherical, what kind of range can be used that will also be efficient enough?
Well, the answer is of course, compounding each of the objects within their own bounding spheres. If the spheres don't overlap, it is certain that there is no collision. If the spheres overlap, there is the possibility of a collision and tests must be performed. It is quite simple creating a bounding sphere for each object and since a sphere is always the same when rotated (invariant when rotated), calculation of the bounding sphere and the object's geometrical center can be done at the initialization stage, usually when objects are loaded.
Calculating the geometric center is exactly finding the average of all the vertices that comprise an object:
This relation gives the X coordinate of the object's center point by summing all X coordinates of all vertices in the object. Notice that in this relation n is NOT the number of faces but the number of vertices of the object. The same calculation can be done for the Y and Z coordinates. The weight functions W can be used to give different weights to vertices but in order to find the geometric center it is usually sufficient to use 1 for any W of any vertex.
After the center is found it is simply a matter of iterating the vertices again and finding the farthest vertex from the center and using that distance as the compound radius.
Step 2 - Finding the closest k faces of object A relative to B The first step in the OFG algorithm is to find the closest geometry object A has in relation to object B. In order to do this in an efficient manner, one very important assumption must be made:
Object B can be considered as a point object located at its center (object centers have been shown in step 1).
This assumption is not a simple one, there is a lot of reasoning behind it. Mainly, in order to find the closest faces object A has relative to B, it is obvious that distances of faces must be considered. The problem is that finding the smallest distances between every face in A and every face in B has a time complexity of O(nm), just as the easiest brute force method presented as the general case. Of course, this will not do.
The idea is to attempt to find distances of all faces in object A, relative to only one geometry property in B. That way, checking every face in A against only one geometry property of object B yields a time complexity of O(n), where n is the number of faces in object A. The question then arises: what kind of geometric property an entire object has that can be used to represent the entire object?
Physicists sometimes assume very distant objects are a point object when trying to simplify problems in physics because when an object is very far, the contribution from any irregularities in the object's geometry are negligible.
For our purposes, this assumption is correct for any distance, mainly due to the fact that this algorithm deals strictly with convex 3D objects. For that reason, the closest faces an object can have to another object are the faces closest to the other object's center.
It is always true that if a certain face is closer to object B's center than another face, it is generally closer to object B than the other face.
Taking the 3 vertices (or more, depending on the engine involved) that constitute a face, it is possible to find that face's center in exactly the same way as finding an object's center. Using the center coordinate coupled with the two object's positions it is possible to automatically calculate a vector from the center of any face to the center of the other object.
In summary, the steps necessary to find the closest k faces of object A are:
Using both object's positions, calculate the relative position vector. Assuming is the center of object A (x,y,z - a 3D vector) and the center of object B, the vector connecting the center points is then:
Start looping through all faces of object A, find each face center relative to object A's center. Total time complexity is O(n). It takes less operations to calculate the faces centers than to do it in the beginning and apply transformations on them.
For each face, find the vector connecting its center with object B's center, using the simple relation: , where is the vector connecting the center of face i with the center of object B, the relative positions of the two object's centers and is the position of the face relative to object A's center (the object it belongs to). Of course, i = 0,1,2,3...,n.
For each , find the vector's size: and store into selection*. Of course, the size itself is irrelevant, as is the direction of the vector. The only important quantity is the vector's size squared. Note that there is no need to take the square root because if a certain face's squared vector size is larger than another face's squared vector size, the same holds for the actual sizes themselves.Figure: Finding object A face vectors relative to object B's center
Step 3 - Calculate the geometric center of the selection and its bounding sphere After finding the closest k faces of object A relative to B it is important to be able to do quick tests in order to check the possibility of a collision between faces. For that reason and another reason outlined in step 4, the center of the new selection must be found and its bounding sphere.
Assuming the previous step has found the necessary k faces, finding the center is as simple as finding the average of all the selected faces centers. Remember, the selection itself is just a means to remember which faces are the closest, and each face has its center coordinate and a vector connecting the center of the face with the center of object B. Therefore we have:
Where is of course the center of the new selection and is the position of the i'th face center relative to the object's center (as always). The resulting vector is the position of the new selection center relative to our object's (A) center.
Once the center is found, the farthest vertex from the new selection center gives the bounding sphere radius. This is a preparation phase for a later step where the possibility of face collision should be tested.
Step 4 - Finding the closest k faces of object B relative to the new selection This part of the algorithm is very similar to step 2 in that it finds the closest faces in B relative to some point. However in this case the point that distances are calculated to is NOT object A's center. Rather, taking the center of the new selection found in step 2 and calculated in step 3 is better. True, for truly convex objects such as spheres there is no difference. Finding the faces in B that are closest to the generally closest faces in A yields better results. Not only is that more accurate but it helps the algorithm deal with objects that are not truly convex but only close to being convex.
Since this step is so similar to step 2, only the summary of the steps needed is presented:
Calculate the relative position vector between object B's center and the center of the new selection of k faces found earlier. Assuming is the center of object B and the center of the selection in object A, the vector connecting the center points is then:
Start looping through all faces of object B, find each face center relative to the center of the selection in object A. Total time complexity is O(m). It takes less operations to calculate the faces centers than to do it in the beginning and apply transformations on them.
For each face, find the vector connecting its center with the center of the selection in object A, using the simple relation: , where is the vector connecting the center of face i with the center of the selection in object A, the relative positions of the center of object B and the center of the selection in object A and is the position of the face relative to object B's center (the object it belongs to). Of course, i = 0,1,2,3...,m.
For each , find the vector's size: and store into selection*.Step 5 - Calculate the geometric center of the new selection and its bounding sphere In essence, step 5 is identical to step 3 only it operates on the newly selected faces in object B. Averaging the faces with the following relation:
Will give the center of the newly selected faces in object B relative to B's center naturally. Once the bounding sphere radii of both the selection from object A and B are known it is a simple matter to accomplish the next step, step 6.
Step 6 - Test collision between the two selections' bounding spheres The last step before any accurate collision tests is the bounding sphere test for the two selections. Up until now the only test done is the bounding sphere test for the two objects that determine if there is a chance for a collision. Now that there are two sets of faces that have a bounding sphere, it is a simple matter of testing whether or not the spheres intersect in order to determine whether real geometry tests should be performed.
Denoting with the center of the selection in object A and the center of the selection in object B, the radii as and respectively, in order to determine if there is intersection the following relation holds:
What this relation means is that if the size of the vector that connects the centers, squared, is smaller/equal to the sum of the radii, squared, there is the possibility of a collision between faces in the selections. Actually, the real check is against the square roots of both expressions but it holds for the squares too. There is no need to waste valuable processing time in order to perform two square root operations. The size of the vector is naturally a dot product of it with itself.
Step 7 - Sort the two selections in ascending order This step can be omitted if wanted but it can help gain a small performance boost. Assuming two selections exist with k elements in each (the elements being the faces to be checked), this step just sorts each of the selections, from the face with the smallest distance to the face with the largest distance. The order of each sorting operation can vary depending on the algorithm and the insertion technique used earlier when building the selection sets. If the selections are almost sorted, even a simple algorithm such as bubble sort can work in a reasonable time (bubblesort works at O(k) for almost sorted arrays).
The algorithm to choose from is really up to the programmers implementing this step. It depends entirely on the insertion method used earlier and there are several good sorting algorithms.
Step 8 - Perform intersection tests on the two selection sets If the algorithm made it this far, it is now time to examine the geometry itself for collision. As input, there are two selections of k faces each, the closest faces between the two objects. Under the assumptions made earlier, the problem of testing for intersection of a pair of faces is a well solved problem (again, see the appendix for ways of doing this), therefore in this subsection a summary of the testing method is described. As mentioned earlier, k can be between 4 and 8.
It is possible to examine all the faces in one set against all the faces in the other set. After all, there are k faces in each set and there are two sets, thus the worst case time complexity is O(k2). This holds true no matter what but certain optimizations on the order of the tests can make a difference.
Problems with the basic OFG algorithm The OFG algorithm suffers from three very serious problems:
The algorithm supports detection of mostly convex objects. Some concave objects will work as well but there are bound to be degenerate cases that cause the algorithm to fail. It hasn't been proven mathematically that there are any cases that will cause failure but assuming there aren't is not a good idea.
Moving objects in computer simulations are by nature problematic because time in computer systems is discrete. In numerical approximations, time is discrete and therefore object positions are calculated in time steps and are really "teleporting" in small steps to create the illusion of motion. The problem arises when the velocity of objects is very large and/or objects are very small. It is quite possible that an object will be close to another object while not colliding, yet at the next time step will be half intersecting with the other object. The problem is then what faces to test. After all, the "closest" faces the object previously had are now inside the other close object. Even more so, it's possible that if the velocity is large enough, the object might pass right through the other object without any way of us detecting this.
In a similar fashion, rotating objects pose another problem. If objects have an angular velocity (or momentum, whichever you prefer) it's possible that before the collision some faces are the closest selection while after a small time step, other faces should be the closest selection. For example, consider a normal cube floating above a table at a small height. The closest faces to the table are naturally the two faces (triangles) that comprise the base of the cube. If the cube is about to rotate 45 degrees in one time step (very fast rotation) it wouldn't be correct checking the base of the cube against the table.
In the following subsections some solutions are presented that deal with said problems.
Solving the concavity problem Since some objects are almost convex and some are not even close to being convex, a method is required that can handle these objects. It just so happens that concave objects can be represented by a collection of convex objects. This is implied since any object can be approximated by triangles, which are convex polygons. If there is any doubt about whether an object will cause problems with the OFG algorithm, it is best to represent it using two or more convex objects.
This presents another slight difficulty: if an object is really a collection of objects, which object's geometry is used when building the selection set?
Well, the solution to this lies in the OFG method itself, but at the object level. Just as faces, it is possible to generalize the algorithm for objects. For example, if two concave objects exists that are made out of an assortment of convex objects each, consider the following scheme:
Find the closest object in collection A relative to the entire collection B. This implies each object in A has to have a center (this is mandatory for OFG anyway) and the entire collection has to have a center as well (averaging the centers of the objects, should be a part of initialization).
Find the closest object in collection B relative to the object found earlier in collection A.
Feed the two objects found to the OFG algorithm.
Solving the discrete time problem A technique exists that makes this very easy to accomplish. Consider a 2D face passing through a wall straight to the other side. This is only possible in a computer simulation if the face is moving fast enough. If it is, it can find itself on the other side of the wall after a small time unit.
Consider connecting each vertex that comprise the face with the same vertex on an exact copy of the face but on the other side of the wall. Not only that, the copy face's position is in fact the position of the original face at the next time step (after one time unit has elapsed, this is easy to calculate). With this in mind, each pair of "connected" vertices make up an edge. Then, testing the new virtual edges created for intersection with the other object (in this case, the wall) will determine if there is a collision.
Actually, this is not enough. The following steps are in order:
Calculate the translation of object A after a time unit elapses.
Feed the two objects into the OFG algorithm, forgoing step 6 completely and without the bounding sphere calculations in steps 3 and 5. Also, there is no need to perform step 8 (the final step) of the OFG algorithm just yet.
Connect each vertex in selection A with the same vertex on the virtual selection of A that is translated by the amount calculated in step 1. This gives a vector that represents the edge to test against object B.
Test the newly created edges against the selection in object B found by the OFG algorithm. They are the most likely to be intersecting object B.
Test the original faces in selection A against the selection in B, as was intended in step 8 of the OFG algorithm.
Test the translated virtual selection of A against the selection of B using step 8 of the OFG algorithm.
Naturally if any collision is found the entire process is stopped and action is taken. However, since objects are "teleporting", it isn't clear which faces should have collided provided time was not discrete. Figuring out which faces collided will help determine what action to take.
There are generally two ways of solving this, others may exists:
Using the translation found in step 1 earlier and given that it's easily possible to calculate the relative velocity of the two objects, it's possible to estimate the time of collision (therefore obtaining the delta time needed for the objects to collide).
Moving half of the distance (half a time unit) and checking for collisions. If collision occurred, great. If not, move another half, so on and so forth. This has a time complexity of log(n) times of performing all the collision tests.
Figure: One face of the selection in A translated after one time unit.
These two methods should only be used in case a collision occurred in between time steps. If a collision occurred with the original selection or the virtual selection (the beginning position and the ending position respectively), there is no need to interpolate the collision time since the intersecting faces are known.
Solving for rotating objects For the general case of rotating objects, the same approach as in the discrete time solution can be applied. Instead of only translating the vertices and using them to create virtual edges, rotating and then translating the vertices solves the problem. Instead of having the selection in its starting position and ending position, have the virtual selection rotated before translation so it'll be rotated at its new position. Vertices are connected in exactly the same way and figuring the moment of collision if one occurred works in exactly the same way but with rotation in mind.
Appendix A - Detecting collision between two faces Up until now the problem of testing for collision between a pair of faces was a well solved problem. That meant that the algorithm assumed detecting if two faces intersect is a black box operation, the details weren't important for the algorithm itself. Still, in order to create a solid implementation of any collision detection scheme, the problem of intersecting faces must be solved. In this section two methods are presented that attempt this.
The first method is called the intersection-based collision detection and is basically an accurate way of detecting whether any two faces intersect each other. Other methods exist as well, however, the second method presented here is a hybrid method that incorporates bounding spheres and some basic geometry testing.
Intersection-based collision detection The problem of collision detection between faces can be broken down to two stages. A well known fact is that any two planes that are parallel don't intersect each other. A plane of course is a flat, infinitely thin, infinitely long surface in 3D space. Unless the planes that contain the faces are parallel, those two planes are going to intersect each other. The first stage would be to detect if any edges of the first face intersect the plane of the second face. That ensures that the first face at least intersects the plane that contains the second face, but it is not sufficient in order to determine if the first face intersects the actual second face. This will be dealt with in the second stage.
In order to determine if any edges in the first face pierce the second face's plane, the vertices making up the edges must be examined. For simplicity, if two points that define an edge fall on one side of the plane (containing the second face), the edge does not pierce that plane. Then, generalizing for the entire face, if all edges don't pierce the plane, the face does not pierce the plane. If any edge pierces the plane it is assured that at least one edge pierces the plane. For convex polygons there are exactly two edges piercing the plane.
The only thing left to solve is how to find whether an edge pierces the plane. It so happens that all points in space that satisfy the following equation fall on the plane:
This might be familiar to you. It is called the plane equation. Any point that lies on the plane itself will evaluate the equation to be true. It so happens that any points on one "side" of the plane produce a positive sign (instead of 0) and all the points on the other "side" produce a negative sign. Therefore, if an edge is defined by two points (vertices) and solving for both vertices gives the same sign, the edge is definitely not piercing the plane. The logical extension to this is to check all edges against the plane. The check becomes very simple:
Solve the plane equation of the second face for each vertex in the first face. If all vertices produce the same sign, the first face definitely doesn't pierce the plane in question. Creating the plane equation is easy given two vectors on that plane (three vertices of one face can give two vectors). If any vertex gives the other sign, there is a possibility of collision.
If there was any piercing by any edge, the second stage should be used to actually detect whether the edges actually pierce the second face itself, unlike the plane tested against in the first step. Actually, this step is all that is needed in order to determine if two faces intersect. However, the second stage is much slower in comparison with stage one. Stage one can be used as an optimization to rule out any faces that definitely don't collide.
Because this algorithm deals with convex polygons (faces actually), this stage will assume the same. Since step one stopped when an edge that pierces the second plane was found, and there are actually two such edges (again, for convex polygons such as triangles), there must be exactly two intersection points between the face and the plane of the other face. Using the plane equation and solving for the x,y and z coordinates of the point that represent the intersection between the plane and the edge it is possible to find those two intersection points. Those points are naturally on the plane itself.
Once two intersection points are found, in order to determine whether the edge pierced the second face itself, at least one of the points must be within the second face!
What is left then is to determine whether a point is within a convex polygon or not. If one of them is within it, there is a definite collision. If both are not within it, there is definitely no collision. There is one very simple solution to this problem, called the half-space method. The halfspace method is a method to determine if a point lies within a convex polygon. For 2D polygons this is simple. Using the line equations of the edges and solving for the point gives either 0, a negative sign or a positive sign, just like in the previous step. However, our edges are vectors in 3D space so this doesn't apply here.
What can be done instead is create a plane that is perpendicular both to the normal of the face and the edge vector. That plane divides space into two parts and therefore the point must lie either on it, or on one of its sides. The same logic from the previous stage applies here as well:
Calculate the intersection points of the edges found piercing the plane in stage one using the second face's plane equation.
For each edge in the second face, take the cross product of the edge vector with the normal to the face, giving a perpendicular plane.
The points must lie either on the plane or on either side. Using the same logic as in stage one can determine this for each edge plane.
If the point lies in the same part of space ("side") for all planes corresponding to the edges of the face, the point is within the face and therefore a collision has occurred
If both points lie outside the planes of the edges (not all signs are the same for each point checked against the edge planes), there is definitely no collision.
Hybrid collision detection between two faces This method is a proposed method that approximates intersection between a pair of faces. The idea is to find a bounding sphere for both faces. If the bounding spheres intersect (an easy and fast check), there is a need for a better check. If the bounding spheres do not intersect, there is definitely no intersection.
For intersecting bounding spheres, there is the possibility of collision. Consider using the logic in the previous method's stage one to determine whether edges on the first face pierce the second. Although this alone does not guarantee collision of course, coupled with the bounding spheres check it gives a reasonable chance for collision. That is, if the spheres collided AND an edge was found to be piercing the other face, most chances there is a collision. Of course, this method is only an approximation and will not give accurate results such as those provided by the intersection method. However, this method is by far much faster than the intersection based method.
Generally, the smaller the faces, the better this method works. This is because when the faces are smaller, there is less and less "free space" within the sphere. Less free space that can generate a collision between spheres but might not actually generate a collision between the faces themselves.
Appendix B - Determining k, the accuracy constant The reasoning behind assigning a proper k value are totally up to the engine in question. In most cases in a normal 3D surface, each vertex will share a maximum of four polygons (each made of two triangles). Of course, there are cases, such as the tip of a pyramid with n sides that do not satisfy this condition. For the first case, a normal 3D surface, each vertex can be shared by 8 triangles at most and 4 at the very minimum. Therefore those values are chosen as the default accuracy range for most purposes. Assigning a different value might serve different types of geometry better though, it has to be considered carefully.
For the cases where a vertex does not share only 8 triangles (such as the tip of an n-sided pyramid), it is a definite fact that if the vertex falls within another 3D object, all of the n-sides of the pyramid will intersect the other object. Therefore, any value for k that is one or more will suffice for this type of degenerate case. It happens that any 3D geometry can be represented either by the former representation or the latter. The latter has no bearing on the assignment of k. The former has and it has been shown that a value of 6 is the best average while 8 should give good accuracy.
About this document... Opposing Face Geometry A Collision Detection Optimization Scheme This document was generated using the LaTeX2HTML translator Version 2002 (1.62)
So, what is a Quaternion?
Quaternions aren't actually as scary as they sound. Everything I read regarding quaternions talked about imaginary numbers, hyper-complex numbers, spinors, and other scary sounding things. There was too much maths jargon for my liking. If you understand what a vector is, it isn't very hard to understand what a quaternion is. One way to represent a quaternion is
Q = xi + yj + zk + w
where i,j & k are coordinate basis vectors for three dimensions.
I don't particularly like this representation for computer graphics, especially where cameras are concerned. I prefer thinking of a quaternion as an object that contains a vector and a scalar. We'll call the vector v and keep the scalar as w.
Q = [ w, [b]v[/b] ]
where v = xi + yj + zk.
For use in the sample code below, here's a quaternion data structure:
struct quaternion
{
double x, y, z, w;
};
This is a much easier representation to comprehend for me. Now for our purposes, quaternion addition, subtraction, etc., aren't needed. We only need to know how to normalize (scale to length=1), multiply and compute the conjugate of a quaternion in order to generate a rotation. These tasks are actually very simple and are described below.
Normalizing a quaternion
Normalizing a quaternion is almost the same as normalizing a vector. We start by computing the length of the quaternion to be normalized, which is done using the Euclidean distance forumula:
The last thing you need to know is quaternion multiplication. Then you'll be ready to make a quaternion based camera.
Multiplying quaternions
Multiplication with quaternions is a little complicated as it involves dot-products cross-products. However, if you just use the following forumula that expands these operations, it isn't too hard. To multiply quaternion A by quaternion B, just do the following:
That's not too that hard is it? Now we'll look at how to use these operations to make a quaternion based camera.
The quaternion camera
To make a camera you typically use three vectors: Position, View, and Up (or you may call them what you like). For a first person camera - which we will be using - we're only going to consider rotating the View vector. With quaternions we can rotate a vector around an arbitrary axis (same as with axis-angles) very easily.
To achieve this, first we need to turn our View vector into a quaternion, then define a rotation quaternion and lastly, apply the rotation quaternion to the View quaternion to make the rotation.
To make the View quaternion, V, the x, y, and z values are taken from the View vector and we simply add a 0 for the scalar component w. Thus,
V = [0, View]
Then you need to make a quaternion to represent the rotation. To do this, you need the vector you want to rotate about, and the angle you wish to rotate by. We'll just simply term the vector to rotate about A, and the angle theta. Here is the formula to build your rotation quaternion, which we'll call R.
So now we have the vector (View) and its quaternion V that we want to rotate by an angle theta about the vector A. The rotation quaternion R defines this rotation. After the rotation, we'll have the new quaternion representing our view, given by W. The rotation operation is simply
W = R * V * R'
where R' is the conjugate of R. To get our new view vector, we just take the vector components out of W.
NewView = [W.x W.y W.z]
The following function (using SDL, use glut or whatever you like) sets the view based on the distance from the current mouse coordinates to the centre of the screen. I learned how to do this from gametutorials.com, and modified the code for my purposes. In this code, positive x is to the right and positive y is down the screen.
void Camera::SetViewByMouse(void)
{
// the coordinates of our mouse coordinates
int MouseX, MouseY;
// the middle of the screen in the x direction
int MiddleX = SCREENWIDTH/2;
// the middle of the screen in the y direction
int MiddleY = SCREENHEIGHT/2;
// vector that describes mouseposition - center
Vector MouseDirection(0, 0, 0);
// static variable to store the rotation about the x-axis, since
// we want to limit how far up or down we can look.
// We don't need to cap the rotation about the y-axis as we
// want to be able to turn around 360 degrees
static double CurrentRotationAboutX = 0.0;
// The maximum angle we can look up or down, in radians
double maxAngle = 1;
// This function gets the position of the mouse
SDL_GetMouseState(&MouseX, &MouseY);
// if the mouse hasn't moved, return without doing
// anything to our view
if((MouseX == MiddleX) && (MouseY == MiddleY))
return;
// otherwise move the mouse back to the middle of the screen
SDL_WarpMouse(MiddleX, MiddleY);
// get the distance and direction the mouse moved in x (in
// pixels). We can't use the actual number of pixels in radians,
// as only six pixels would cause a full 360 degree rotation.
// So we use a mousesensitivity variable that can be changed to
// vary how many radians we want to turn in the x-direction for
// a given mouse movement distance
// We have to remember that positive rotation is counter-clockwise.
// Moving the mouse down is a negative rotation about the x axis
// Moving the mouse right is a negative rotation about the y axis
MouseDirection.x = (MiddleX - MouseX)/MouseSensitivity;
MouseDirection.y = (MiddleY - MouseY)/MouseSensitivity;
CurrentRotationX += MouseDirection.y;
// We don't want to rotate up more than one radian, so we cap it.
if(CurrentRotationX > 1)
{
CurrentRotationX = 1;
return;
}
// We don't want to rotate down more than one radian, so we cap it.
if(CurrentRotationX < -1)
{
CurrentRotationX = -1;
return;
}
else
{
// get the axis to rotate around the x-axis.
Vector Axis = CrossProduct(View - Position, Up);
// To be able to use the quaternion conjugate, the axis to
// rotate around must be normalized.
Axis = Normalize(Axis);
// Rotate around the y axis
RotateCamera(MouseDirection.y, Axis.x, Axis.y, Axis.z);
// Rotate around the x axis
RotateCamera(MouseDirection.x, 0, 1, 0);
}
}
This function actually rotates our view. After we are done, just plug your camera vectors (Position, View, and Up) into gluLookAt(Position.x, Position.y, Position.z, View.x, View.y, View.z, Up.x, Up.y, Up.z). Here is the code for the rotation.
and your camera should work just perfectly. Maybe some other time, I'll do a third person camera tutorial, and explain how to use SLERP.]]>Mon, 22 Sep 2003 00:58:47 +0000e33d974aae13e4d877477d51d8bafdc4Beat Detection Algorithms
Disclaimer
This document is to be distributed for free and without any modification from its original state. The author declines all responsibility in the damage this document or any of the things you will do with it might do to anyone or to anything. This document and any of its contents is not copyrighted and is free of all rights, you may thus use it, modify it or destroy it without breaking any international law. However according to the author's will, you may not use this document for commercial profit directly, but you may use indirectly its intellectual contents; in which case I would be pleased to receive a mail of notice or even thanks. This is my first tutorial and I am still a student, you must assume that this document is probably not free of small errors and bugs. In the same state of mind, those algorithms are not fully optimised, they are explained for pedagogical purposes and you may find some redundant computations or other voluntary clumsiness. Please be indulgent and self criticise everything you might read. Hopefully, lots of this stuff was taken in sources and books of reference; as for the stuff I did: it has proven some true efficiency in test programs I made and which work as wanted. As said in the introduction: If you have any question or any comment about this text, please send it to the above email address, I'll be happy to answer as soon as possible.
Introduction
Simulating a physical phenomena which obeys to known mathematical equations is, with a number of approximations, always feasable. But what about more abstract concepts, such as feelings, which do not follow any laws? The simplest things we can feel are often the hardest things to capture in a program. Beat detection follows this rule : feeling the beat of a song comes naturally to humans or animals. Indeed it is only a feeling one gets when listening to a melody, a feeling which will make you dance in rhythm or hit a table with your hands on the melody beats. Therefore, how can we teach this beat detection to a machine that can only compute logical operations? In fact there are a number of algorithms which manage to approximate, more or less accurately, this beat detection. We will first study the statistical approach of beat detection on a streaming source and secondly a filtering approach of rhythm extraction on a static song.
This guide assumes the reader has basic signal processing understanding (FFT, convolutions and correlations should sound common) maybe some stuff in statistics will also help (Variance, Average, Principal Components Analysis, will be quoted among others). The point here is not to actually write the code of these algorithms, but more to understand how they work and to be able to adapt or create the appropriate algorithm to a situation. If you have a question or a comment about this text, please send it to the above email address, I'll be happy to answer as soon as possible. Anyway, the aim here is to give more precise ideas on the subject of beat detection to the reader. Enjoy.
I – Statistical streaming beat detection
1 – Simple sound energy
a - A first analysis
The human listening system determines the rhythm of music by detecting a pseudo – periodical succession of beats. The signal which is intercepted by the ear contains a certain energy, this energy is converted into an electrical signal which the brain interprets. Obviously, The more energy the sound transports, the louder the sound will seem. But a sound will be heard as a beat only if his energy is largely superior to the sound's energy history, that is to say if the brain detects a brutal variation in sound energy. Therefore if the ear intercepts a monotonous sound with sometimes big energy peaks it will detect beats, however, if you play a continuous loud sound you will not perceive any beats. Thus, the beats are big variations of sound energy. This first analysis will bring us to our simplest model : Sound energy peaks.
In this model we will detect sound energy variations by computing the average sound energy of the signal and comparing it to the instant sound energy. Lets say we are working in stereo mode with two lists of values : (an) and (bn). (an) contains the list of sound amplitude values captured every Te seconds for the left channel, (bn) the list of sound amplitude values captured every Te seconds for the right channel. So we want to compute the instant energy and the average energy of the signal. The instant energy will in fact be the energy contained in 1024 samples (1024 values of a[n] and b[n]), 1024 samples represent about 5 hundreds of second which is pretty much 'instant'. The average energy should not be computed on the entire song, some songs have both intense passages and more calm parts. The instant energy must be compared to the nearby average energy, for example if a song has an intense ending, the energy contained in this ending shouldn't influence the beat detection at the beginning. We detect a beat only when the energy is superior to a local energy average. Thus we will compute the average energy on say : 44032 samples which is about 1 second, that is to say we will assume that the hearing system only remembers of 1 second of song to detect beat. This 1 second time (44032 samples) is what we could call the human ear energy persistence model; it is a compromise between being to big and taking into account too far away energies, and being too small and becoming to close to the instant energy to make a valuable comparison.
The history buffer where we will keep the last 44032 samples wil contain in fact too lists of samples (B[0]) and (B[1]) corresponding to the left (an) and to the right (bn) channels history.
Simple sound energy algorithm #1:
Every 1024 samples:
Use the 1024 new samples taken in a[n] and b[n] to compute the instant energy 'e', using the following formula (i0 is the position of the 1024 samples to process): (R1)
Compute the local average energy '<E>' on the 44100 samples of a history buffer (B): (R2)
Shift the 44032 history buffer (B) of 1024 indexes to the right so that we make room for the 1024 new samples and evacuate the oldest 1024 samples.
Move the 1024 new samples on top of the history buffer.
Compare 'e' to 'C * ' where C is a constant which will determine the sensibility of the algorithm to beats. A good value for this constant is 1.3. If 'e' is superior to 'C * ' then we have a beat!
b - Some direct optimisations
This was the basic version of the algorithm, its speed and accurecy can be improved quite easily. The algorithm can be optimised by keeping the energy values computed on 1024 samples in history instead of the samples themselves, so that we don't have to compute the average energy on the 44100 samples buffer (B) but on the instant energies history we will call (E). This sound energy history buffer (E) must correspond to approximately 1 second of music, that is to say it must contain the energy history of 44032 samples (calculated on groups of 1024) if the sample rate is 44100 samples per second. Thus E[0] will contain the newest energy computed on the newest 1024 samples, and E[42] will contain the oldest energy computed on the oldest 1024 samples. We have 43 energy values in history, each computed on 1024 samples which makes 44032 samples energy history, which is equivalent to 1 second in real time. The count is good. The value of 1 second represents the persistance of the music energy in the human ear, it was obtain with experimentations but it may varry a little from a person to another, just adjust it as you feal. So here is what the algorithm becomes:
Simple sound energy algorithm #2:
Every 1024 samples:
Compute the instant sound energy 'e' on the 1024 new sample values taken in (an) and (bn) using the formula (R1)
Compute the average local energy <E> with (E) sound energy history buffer: (R3<E>'.
c - Sensitivity detection
The imediate draw back of this algorithm is the choice of the 'C' constant. For example in techno and rap music beats are quite intense and precise so 'C' should be quite high (about 1.4); whereas for rock and roll, or hard rock which contains a lot of noise, the beats are more confused and 'C' should be low (about 1.1 or 1.0). There is a way, to make the algorithm determine automatically the good choice for the 'C' constant. We must compute the variance of the energies contained in the energy history buffer (E). This variance, which is nothing but the average of ( Energy Values – Energy average = (E) - <E>), will quantify how marked the beats of the song are and thus will give us a way to compute the value of the 'C' constant. The formula to calculate the variance of the 43 E[i]values is described below (R4). Finally, the greater the variance is the more sensitive the algorithm should be and thus the smaller 'C' will become. We can choose a linear decrease of 'C' with 'V' (the variance) and for example when V 200, C -> 1.0 and when V -> 25, C -> 1.45 (R5). This is our new version of the sound energy beat detection algorithm:
Simple sound energy algorithm #3:
Every 1024 samples:
Compute the instant sound energy 'e' on the 1024 new samples taken in (an) and (bn) using the following formula (R1).
Compute the average local energy <E> with (E) sound energy history buffer using formula (R3).
Compute the variance 'V' of the energies in (E) using the following formula: (R4) Compute the 'C' constant using a linear degression of 'C' with 'V', using a linear regression with values (R5): (R6[lessthanE>', if superior we have a beat!
Those three algorithms were tested with several types of music, among others : pop, rock, metal, techno, rap, classical, punk. The fact is the results are quite unpredictable. I will only talk about Simple beat detection algorithm #3 as #2 and #1 are only pedagogical intermediates to get to the #3.
Clearly, the beat detection is very accurate and sounds right with techno and rap, the beats are very precise and the music contains very few noise. The algorithm is quite satisfying for that kind of music and if you aim to use beat detection on techno you can stop reading here, the rest won't change anything to your beat detection. However, even if the improvement of the dynamic 'C' calculations ameliorates things alot, the beat detection on punk, rock and hard rock, is sometimes quite approximate. We can feel it doesn't really get the rythm of the song. Indeed the algorithm detects energy peaks. Sometimes you can hear a drum beat which is sank among other noises and which goes trough the algorithm without being detected as a beat.
To explain this phenomena lets say a guitare and flute make alternatively an amplitude constant note. Each time the first finishes the other starts. The note made by the guitare and the note made by the flute have the same energy but the ear detects a certain rhythm because the notes of the instruments are at different pitch. For our algorithm (who is one might say colorblind) it is just an amplitude constant noise with no energy peaks. This partly explains why the algorithm doesn't detect precisely beats in songs with a lot of instruments playing at different rythms and simultaneously. Our next analysis will make us walk through this difficulty.
Comparing the results we have obtained with the Simple beat detection algorithm #3 to its computing cost, this algorithm is very efficient. If you are not looking for a perfect beat detection than I recommend you use it. Here is a screenshot of a program I made using this algorithm. You fill find the binaries and the sources on my homepage.
Figure 1: The spectrum analyser is not useful for the beat detection it is only for visual matters, but you can see at the top some of the parameters the program computes to execute the algorithm.
2 – Frequency selected sound energy
a - The idea and the algorithm
The issue with our last analysis of beat detection is that it is colorblind. We have seen that this could raise quite a few problems for noisy like songs in rock or pop music. What we must do is give our algorithm the abbility to determine on which frequency subband we have a beat and if it is powerful enough to take it into account. Basically we will try to detect big sound energy variations in particular frequency subbands, just like in our last analysis; unless this time we will be able to seperate beats regarding their color ( frequency subband ). Thus If we want to give more importants to low frequency beats or to high frequency beats it should be more easy. Notice that the energy computed in the time domain is the same as the energy computed in the frequency domain, so we don't have any difference between computing the energy in time domain or in frequency domain. For maths freaks this is called the Parseval Theorem.
Okay that was just a bit of sport, lets go back to the mainstream; Here is how the Frequency selected sound energy algorithm works: The source signals are still coming from (an) and (bn). (an) and (bn) can be taken from a wave file, or directly from a streaming microphone or line input. Each time we have accumulated 1024 new samples, we will pass to the frequency domain with a Fast Fourier Transform (FFT). We will thus obtain a 1024 frequency spectrum. We then divide this spectrum into however many subbands we like, here I will take 32. The more subbands you have, the more sensitive the algorithm will be but the harder it will become to adapt it to lots of different kinds of songs. Then we compute the sound energy contained in each of the subbands and we compare it to the recent energy average corresponding to this subband. If one or more subbands have an energy superior to their average we have detected a beat.
The great progress with the last algorithm is that we now know more about our beats, and thus we can use this information to change an animation, for example. So here is more precisely the Frequency selected sound energy algorithm #1:
Every 1024 samples:
Compute the FFT on the 1024 new samples taken in (an) and (bn). The FFT inputs a complex numeric signal. We will say (an) is the real part of the signal and (bn) the imagenary part. Thus the FFT will be made on the 1024 complex values of: You can find FFT tutorials and codes in C, Visual Basic or C++ on my homepage in the 'tutorials' section or by typing 'FFT' on Google.
From the FFT we obtain 1024 complex numbers. We compute the square of their module and store it into a new 1024 buffer. This buffer (B) contains the 1024 frequency amplitudes of our signal.
Divide the buffer into 32 subbands, compute the energy on each of these subbands and store it at (Es). Thus (Es) will be 32 sized and Es[i] will contain the energy of subband 'i': (R7)
Now, to each subband 'i' corresponds an energy history buffer called (Ei). This buffer contains the last 43 energy computations for the 'i' subband. We compute the average energy <E> for the 'i' subband simply by using: (R8)
Shift the sound energy history buffers (Ei) of 1 index to the right. We make room for the new energy value of the subband 'i' and flush the oldest.
Pile in the new energy value of subband 'i' : Es[i] at Ei[0]. (R9)
For each subband 'i' if Es[i] > (C*<E>) we have a beat !
To help out visualising how the data piles work have a look at this scheme:
Figure 2: This is how the energy data is organized.
Now the 'C' constant of this algorithm has nothing to do with the 'C' of the first algorithm, because we deal here with separated subbands the energy varies globally much more than with colorblind algorithms. Thus 'C' must be about 250. The results of this algorithm are convincing, it detects for example a symbal rhythm among other heavy noises in metal rock, and indeed the algorithm separates the signal into subbands, therefore the symbal rhythm cannot pass trough the algorithm without being recognized because it is isolated in the frequency domain from other sounds. However the complexity of the algorithm makes it useful only if you are dealing with very noisy sounds, in other cases, Simple beat detection algorithm #3 will do the job.
b - Enhancements and beat decision factors.
There are ways to enhance a bit more our Frequency selected sound energy algorithm #1.
First we will increase the number of subbands from 32 to 64. This will take obviously more computing time but it will also give us more precision in our beat detection. The second way to develop the accuracy of the algorithm uses the defaults of human ears. Human hearing system is not perfect; in fact its transfer function is more like a low pass filter. We hear more easily and more clearly low pitched noises than high pitch noises. This is why it is preferable to make a logarithmic repartition of the subbands. That is to say that subband 0 will contain only say 2 frequencies whereas the last subband, will contain say 20. More precisely the width 'wi' of the 'n' subbands indexed 'i' can be obtained using this argument:
Linear increase of the width of the subband with its index: (R10)
We can choose for example the width of the first subband: (R11)
The sum of all the widths must not exceed 1024 ((B)'s size): (R12)
Once you have equations (R11) and (R12) it is fairly easy to extract 'a' and 'b', and thus to find the law of the 'wi'. This calculus of 'a' and 'b' must be made manually and 'a' and 'b' defined as constants in the source; indeed they do not vary during the song.
So in fact in Frequency selected sound energy algorithm #1, all we have to modify is the number of subbands we will take equal to 64 and the (R7) relation. This relation becomes:
(R7)'
It may seem rather complicated but in fact it is not. Replacing this relation (R7) with (R7)' we have created Frequency selected sound energy algorithm #2. If you have musics with very tight and rapid beats, you may want to compute the stuff more frequently than every 1024 samples, but this is only for special cases, normally the beat should not be shorter than 1/40 of second.
Using the advantages of Frequency selected beat detection you can also enhance the beat decision factor. Up to now it was based on a simple comparison between the instant energy of the subband and the average energy of the subband. This algorithm enables you to decide beats differently. You may want for examples to cut beats which correspond to high pitch sounds if you run techno music or you may want to keep only [50-4000Hz] beats if you are working with speech signal. This algorithm has the advantage of being perfectly adaptable to any kind or category of signal which was not the case of Simple beat detection algorithm #3. Notice that the correspondants between index 'i' of the FFT transform and real frequency is given by formula:
If 'i' < 'N/2' then: (R13)
Else 'i' >= 'N/2' then: (R14)
So 'i' is the index of the value in the FFT output buffer, N is the size of the FFT transform (here 1024), fe is the sample frequency (here 44100). Thus index 256 corresponds to 10025 Hz. This formula may be useful if you want to create your own subbands and you want to know what the correspondants between indexes and real frequency are.
Another way of filtering the beats, or selecting them, is choosing only those which are marked and precise enough. As we have seen before, to detect the accuracy of beats we must compute the variance of the energy histories for each subband. If this variance is high it means that we have great differences of energy and thus that the beat are very intense. Thus all we have to do is compute this variance for each subband and add a test in the beat detection decision. To the "Es[i] > (C*<Ei>)" condition we will add "and V((Ei))>V0". V0 will be the variance limit, with experience 150 is a reasonable value. Now the V((Ei)) value is easy to compute, just follow the following equality if you don't see how:
(R15)
The last (finally) way to enhance your beat detection, is to make the source signal pass through a derivation filter. Indeed differentiating the signal makes big variations of amplitude more marked and thus makes energy variations more marked and recognisable later in the algorithm. I haven't tried this optimisation but according to some sources this is quite useful. If you try it please give me your opinion on it!
Concerning the results of the Frequency selected sound energy algorithm #2 I must admit they are way more satisfying than the Simple sound energy algorithm #3. In a song the algorithm catches the bass beats as well as the tight cymbals hits. I insist on the fact that you may also select the beats in very different ways, which becomes quite useful if you know you are going to run techno music with low pitch beats for example. You may also select the beats differently according to there accuracy with the variance criteria. There are many other ways to decide beats; it is up to you to explore them and find the one which fits the most your needs.
I used Frequency selected sound energy algorithm #2 algorithm in a demo program of which you can see some screenshots just below. One can see quite clearly that there is a beat in low frequencies (probably a bass or drum hit) and also a high pitch beat (probably a cymbal or such):
Figure 3: The histogram represents the instant energy of the 128 subbands.
Figure 4: On this screenshot you can see the 128 subbands E/<E> ratios, and the bandwidth of the subbands. When the algorithm detects a beat in a subband, it overwrites the ratio in red.
The reason why I called this part of the document, 'Statistical streaming beat detection', is that the energy can be considered as a random variable, of which we have been calculating values over time. Thus we could interpret those values as a sampling for the statistical analysis of a random variable. But one can push this approach further. When we have separated the energies histories into 128 subbands we created in fact 128 random energy variables. We can then apply some of the general statistical methods of analysis. For example, the principal components analysis method will enable you to determine if some of the subbands are directly linked or independent. This would help us to regroup some subbands which are directly linked and thus make the beat decision more efficient. However, this method is basically just far too computing expensive and maybe just too hard to implement comparing to the results we want. If you are looking for a really good challenge in beat detection you could push in this direction.
II – Filtering rhythm detection
1 – Derivation and Comb filters
a - Intercorrelation and train of impulses
While in the first part of this document we had seen beat detection as a statistical sound energy problem, we will approach it here as a signal processing issue. I haven't tried myself this algorithm but I greatly inspired myself of some work which was done and tested with this algorithm (see the sources section), so it really works but I won't detail so much its implementation as I did in the first part. The basic idea is the following. If we have two signals x(t) and y(t), we can evaluate a function which will give us an idea of how much those two signals are similar. This function called 'inter correlation function' is the following:
(R16)
For our purposes we will always take alpha=1. This function quantifies the energy that can be exchanged between the signal x(t) and the signal y(t – b), it gives an evaluation of how much those two functions look like each other. The role of the 'b' is to eliminate the time origin problem; two signals should be compared without regarding their possible time axis translation. As you may have noticed, we are creating this algorithm for a computer to execute; basically we have to re-write this formula in discrete mode, it becomes:
(R17)
Now the point is we can valuate the similarity of our song and of a train of impulses using this function. The train of impulses being at a known frequency (the beats per minute we want to test) is what we call a comb filter. The energy of the Y function gives us an evaluation of the similarity between our song and this train of impulses, thus it quantifies how much the rhythm of the train of impulses is present in the song. If we compute the energy of the ? function for different train of impulses, we will be able to see for which of them the energy is the biggest, and thus we will know what is the principal Betas Per Minute rate. This is called combfilter processing. We will note 'Ey' this energy, where x[k] is our song signal and y[k] our train of impulses:
(R18)
By the way, here is what a train of impulses looks like:
Figure 5: This is the train of impulses function, it is caracterised by its period Ti.
So the period Ti of the impulses must correspond to the beats per minutes we want to test on our song. The formula that links a beat per minute value with the Ti in discrete time is the following:
(R19)
fs is the sample frequency, if you are using good quality wave files, it is usually of 44100. BPM is the beats per minute rate. Finally, Ti is the number of indexes between each impulse.
Now because it is quite computing expensive to compute the (R17) formula, we pass to the frequency domain with a FFT and compute the product of the X[v] and Y[v] (FFT's of x[k] and y[k]). Then we compute the energy of the ? function directly in the frequency domain, thus Ey is given with by the following formula:
(R17)'
b - The algorithm
Note that this algorithm is much too computing expensive to be ran on a streaming source, or on a song in a whole. It is executed on a part of the song, like 5 seconds taken somewhere in the middle. Thus we assume that the tempo of the song is overall constant and that the middle of the song is tempo-representative of the rest of the song. So finally here are the steps of our algorithm:
Derivation and CombfilterFor all the beats per minute you want to test, for example from 60 BPM to 180 BPM per step of 10, we will note BPMc the current BPM tested:
Finally compute the energy of the correlation between the train of impulses and the 5 seconds signal using (R17)' which becomes: (R20)
Save E(BPMc) in a list.
The rhythm of the song is given by the BPMmax, where the max of all the E(BPMc) is taken. We have the beat rate of the song!
The AmpMax constant which appears in the train of impulse generation, is given by the sample size of the source file. Usually the sample size is 16 bits, for a good quality .wav. If you are working in non-signed mode, the AmpMax value will be of 65535, and more often if you work with 16 bits signed samples the AmpMax will be 32767.
One of the first ameliorations that could be given to Derivation and Combfilter algorithm #1 is indeed adding a derivation filter before the combfilter processing. To make beats more detectable, we differentiate the signal. This accentuates when the sound amplitude changes. So instead of dealing with a[k] and b[k] directly we first transform them using the formula below. This modification added before the second step of Derivation and Combfilter algorithm #1 constitutes Derivation and Combfilter algorithm #2.
(R21)
As I said before, I haven't tested this algorithm myself; I can't compare its results with the algorithms of the first part. However, throughout the researches I made this seems to be the algorithm universally accepted as being the most accurate. It does have disadvantages: Great computing power consumption, can only be done on a small part of a song, assumes the rhythm is constant throughout the song, no possible streaming input (for example from a microphone), the signal needs to be known in advance. However it returns a deeper analysis than the first part algorithms, indeed for each BPM rate you have a value of its importance in the song.
2 – Frequency selected processing
a - Curing colorblindness
As in the first part of the tutorial, we have the same problem our last algorithm doesn't make the difference between cymbals beat or a drum beat, so the beat detection becomes a bit dodgy on very noisy music like hard rock. To heal our algorithm we will proceed as we had done in Frequency selected sound energy algorithm #1 we will separate our source signal into several subbands. Only this time we have much more computing to do on each subband so we will be much more limited in their number. I think that 16 is a good value. We will use a logarithmic repartition of these subbands as we had done before.
Let's modify the Derivation and Combfilter algorithm #2. The values particular to a subband will be characterized with a little 's' at the end of the name of the variable. So we will separate ta[k] and tb[k] into 16 subbands. Each subbands values array will be called tas[k] and tbs[k] ('s' varies from 1 to 16).
Frequency selected processing combfiltersDifferentiate the a[k] and b[k] signals using (R21).Generate the 16 subband array values tas[k] and tbs[k] by cutting ta[k] and tb[k] with a logarithmic rule.
For all the subbands tas[k] and tbs[k] (s goes from 1 to 16). Ws is the length of the 's' subband.
For all the beats per minute you want to test, for example from 60 BPM to 180 BPM per step of 10, we will note BPMc the current BPM tested : Finally compute the energy of the correlation between the train of impulses and the 5 seconds signal using (R17)' which becomes: (R22)
Save E(BPMc,s) in a list.
The rhythm of the subband is given by the BPMmaxs, where the max of all the E(BPMc,s) is taken (s is fixed). We will call this max EBPMmaxs. We have the beat rate of the subband we store it in a list.
We do this for all the subbands, we have a list of BPMmaxs for all subbands, we can than decide which one to consider.
As in Frequency selected sound energy algorithm #2 we will then be able to decide the rhythm according to frequency bandwidth criteria. Or, if you want to take all the subbands into account you can compute the final BPM of the song, by calculating the barycentre of all the BPMmaxs affected with their max of E(BPMc,s). Like this:
(R23)
I must admit I haven't concretely tested this algorithm. Others have done this already and here an overview of the results for Derivation and Combfilter algorithm #2 (source:
Figure 6: This is plot of the EBPMc values function of BPMc. The algorithm will take the max of the EBPMc value to find the final BPM of the song. Here we see this max is reached for BPMc=75. The final BPM is 75!
Conclusion
Finding algorithms for beat detection is very frustrating. It seems so obvious to us, humans, to hear those beats and somehow so hard to formalise it. We managed to approximate more or less accurately and more or less efficiently this beat detection. But the best algorithms are always the ones you make yourself, the ones which are adapted to your problem. The more the situation is precise and defined the easier it is! This guide should be used as a source of ideas. Even if beat detection is far from being a crucial topic in the programming scene, it has the advantage of using lots of signal processing and mathematical concepts. More than an end in itself, for me, beat detection was a way to train to signal processing. I hope you will find some of this stuff useful.
]]>Sat, 07 Jun 2003 20:39:13 +0000b8c78ee23d4f42c6c58cede44fedb0cdVectors and Matrices: A Primer
Note: This article has since been revised and updated from its original published version you see here.
Preface
Hey there! This tutorial is for those who are new to 3D programming, and need to brush up on that math. I will teach you two primary things here, Vectors and Matrices (with determinants). I'm not going to go into everything, so this isn't designed as a standalone reference. A lot of mathematics books can probably discuss this much better, but anyway, without further ado, lets get on with it shall we?
Vectors
Vector basics – What is a vector?
Vectors are the backbone of games. They are the foundation of graphics, physics modelling, and a number of other things. Vectors can be of any dimension, but are most commonly seen in 2 or 3 dimensions. I will focus on 2D and 3D vectors in this text. Vectors are derived from hyper-numbers, a sub-set of hyper-complex numbers. But enough of that, you just want to know how to use them right? Good and I will teach you 2 of them: vector equations and column vectors. Vectors can also be written using the two end points with an arrow above them. So, if you have a vector between the two points A and B, you can write that as .
A vector equation takes the form a=xi + yj + zk
i, j and k are unit vectors in the 3 standard Cartesian directions. i is a unit vector aligned with the x axis, j is a unit vector aligned with the y axis, and k is a unit vector aligned with the z axis. Unit vectors are discussed later.
The coefficients of the i, j and k parts of the equation are the vector's components. These are how long each vector is in each of the 3 axes. This may be easier to understand with the aid of a diagram.
This diagram shows a vector from the origin to the point ( 3, 2, 5 ) in 3D space. The components of this vector are the i, j and k coefficients ( 2, 3 and 5 ). So, in the above example, the vector equation would be:
a = 2i + 3j + 5k
This can also be related to the deltas of a line going through 2 points.
The second way of writing vectors is as column vectors. These are written in the following form
where x, y and z are the components of that vector in the respective directions. These are exactly the same as the components of the vector equation. So in column vector form, the above example could be written as:
There are various advantages to both of the above forms, but I will continue to use the column vector form, as it is easier when it comes to matrices. Position vectors are those that originate from the origin. These can define points in space, relative to the origin.
Vector Math
You can manipulate vectors in various ways, including scalar multiplication, addition, scalar product and vector product. The latter two are extremely useful in 3D applications.
There are a few things you should know before moving to the methods above. The first is finding the modulus (also called the magnitude) of a vector. This is basically its length. This can be easily found using Pythagorean theorem, using the vector components. The modulus of a is written |a|.
in 3D and in 2D, where x, y and z are the components of the vector in the 3 axes of movement. Unit vectors are vectors with a magnitude of 1, so |a| = 1.
Addition
Vector addition is pretty easy. All you do is add the respective components together. So for instance, take the vectors:
The addition of these vectors would be:
Get it? This can also be represented very easily in a diagram, but I will only consider this in 2D, because it's easier to draw. All you do is subtract the components in one vector from the components in the other. The geometric representation however is very different.
Let and
Then
The visual representation of this It may be easier to think of this as a vector addition.
Where instead of having:
c = a – b
we have
c = -b + a
which, according to what was said about the addition of vectors would produce:
You can see that putting a on the end of –b has the same result.
Scalar multiplication
Scalar multiplication is easy to come to grips with. All you do is multiply each component by that scalar.
So, say you had the vector a and a scalar k, you would multiply each component by the scalar, getting this result You can use scalar multiplication to find the unit vector of another vector. So, take the following example:
To find the unit vector of this, we would divide a by |a|. Calling the unit vector "b":
That is the unit vector b in the direction of a. This just scales each of the components, so that the magnitude is equal to 1.
Scalar multiplication is also used in the vector equation discussed earlier. The constants x, y and z are the scalars that scale the i, j and k vectors, before adding them to find the resultant vector.
The Scalar Product (Dot Product)
The scalar product, also known as the dot product, is very useful in 3D graphics applications. The scalar product is written and is read "a dot b".
The definition the scalar product is the following:
Where q is the angle between the 2 vectors a and b. This produces a scalar result, hence the name scalar product. From this you can see that the scalar product of 2 parallel unit vectors is 1, as |a||b| = 1, and cos(0) also is 1. You should also have seen that the scalar product of two perpendicular vectors is 0, as cos(90) = 0, which makes the rest of the expression 0. The geometric interpretation is:
The scalar product can also be written in terms of Cartesian components., I will not go into how this is derived, but the final, simplified formula of a.b is:
We can now put these two equations equal to each other, yielding the equation:
With this, we can find angles between vectors. This is used extensively in the lighting part of the graphics pipeline, as it can see whether a polygon is facing towards or away from the light source. This is also used in deciding what side of planes points are on, which is also used extensively for culling.
The Vector Product (Cross Product)
The vector product, which is also commonly known as the cross product is also useful. The vector product basically finds a vector perpendicular to two other vectors. Great for finding normal vectors to surfaces.
For those that are already familiar with determinants, the vector product is basically the expansion of the following determinant:
For those that aren't, the vector product in expanded form is:
Read "a cross b".
Since the cross product finds the perpendicular vector, we can say that:
i x j = k
j x k = i
k x i = jThis first finds the vector perpendicular to the plane made by a and b then scales that vector so it has a magnitude of 1. Note however, that there are 2 possible normals to the plane defined by a and b. You will get different results by swapping a and b in the vector product. That is:
This is a very important point. If you put the inputs the wrong way round, the graphics API will not produce the desired lighting, as the normal will be facing in the opposite direction and v is the vector. t is called the parameter, and scales v. From this it is easy to see that as t varies, a line is formed in the direction of v. If t only takes positive values, then p0 is the starting point of the line. Diagrammatically, this is:
In expanded form, the equation becomes:
This is called the parametric form of a straight line.
Using this to find the vector equation of a line through two points is easy:
If t is confined to values between 0 and 1, then what you have is a line segment between the points p0 and p1.
Using the vector equation we can define planes, and test for intersections. I won't go into planes much here, as there are many tutorials on them elsewhere, I'll just skim over it.
A plane can be defined as a point on the plane, and two vectors that are parallel to the plane, or:
where s and t the parameters and u and v are the vectors that are parallel to the plane. Using this, it becomes easy to find the intersection of a line and a plane, because the point of intersection must lie on both the line and the plane, so we simply make the two equations equal to each other.
Take the line:
and the plane:
To find the intersection point we simply equate, so that:
Ed. note: The v on the left in the above equation is not the same vector as the v on the right.
We then solve for w, s and t, and then plug into either the line or plane equation to find the point. When testing for a line segment intersection, w must be between 0 and 1.
There are many benefits for using the normal-distance form of a plane too. It's especially useful for testing what sides of a plane points or other shaped objects are. To do this, you dot the normal vector and the position vector of the point being tested, and add the distance of the plane from the origin.
So, if you have the plane
and the point ( x, y, z ), the point is in front of the plane if
and behind if it is < 0. If the result equals zero, the point is on the plane. This is used heavily in culling and BSP trees.
Matrices
What is a Matrix anyway?
A matrix can be considered a 2D array of numbers. They take the form:
Matrices are very powerful, and form the basis of all modern computer graphics, the advantage of them being that they are so fast. We define a matrix with an upper-case bold type letter. Look at the above example
The identity matrix can be any dimension, as long as it is also a square matrix. The zero matrix is a matrix that has all its elements set to 0.
The elements of a matrix are all the numbers in it. They are numbered by the row/column position so that :
Vectors can also be used in column or row matrices. I will use column matrices here so that it is easier to understand. A 3D vector a in matrix form will use a matrix A with dimension 3x1 so that:
which you can see is the same layout as using column vectors.
Matrix arithmetic
I won't go into every matrix manipulation, but instead I'll focus on the ones that are used extensively in computer graphics.
Matrix Multiplication
There are two ways to multiply a matrix: by a scalar, and by another conformable matrix. First, let's deal with the matrix/scalar multiplication.
This is pretty easy, all you do is multiply each element by the scalar. So, let A be the original matrix, B be the matrix after multiplication, and k the constant. We perform:
where i and j are the positions in the matrix.
this can also be written as: So, in general terms:
Take three matrices A, B and C where C is the product of A and B. A and B have dimension mxn and pxq respectively. They are conformable if n=p. The matrix C has dimension mxq.
You perform the multiplication by multiplying each row in A by each column in B. So let A have dimension 3x3 and B have dimension 3x2.
So, with that in mind, let's try an example:
It's as simple as that! Some things to note:
A matrix multiplied by the identity matrix is the same, so:
The transpose of a matrix is it flipped on the diagonal from the top left to the bottom right, so for example:
The transpose of this matrix would be:
Simple enough eh? And you thought it was going to be hard!
Determinants
I'm going to talk a little bit about determinants now, as they are useful for solving certain types of equations. I will discuss easy 2x2 determinants first.
Take a 2x2 matrix:
The determinant of a matrix A is written |A| and is:
For higher dimension matrices, the determinant gets more complicated. Let's discuss a 3x3 matrix. You pass along the first row, and at each element, you discount the row and column that intersects it, and calculate the determinant of the resultant 2x2 matrix multiplied by that value.
So, for example, take this 3x3 matrix:
Ok then, Step 1: move to the first value in the top row, a11 . Take out the row and column that intersects with that value. Step 2: multiply that determinant by a11. So, using diagrams:
Step1:
Step2:
We repeat this all along the top row, with the sign in front of the value of the top row alternating between a "+" and a "-", so the determinant of A would be:
Now, how do we use these for equation solving? Good question. I will first show you how to solve a pair of equations with 2 unknowns.
Take the two equations:
We first push the coefficients of the variables into a determinant, producing:
You can see it's laid out in the same way, which makes it easy. Now, to solve the equation in terms of x, we replace the x coefficients in the determinant with the constants k1 and k2, dividing the result by the original determinant. So, that would be:
To solve for y we replace the y coefficients with the constants instead.
Let's try an example to see this working:
We push the coefficients into a determinant, and solve:
To find x substitute constants into x co-efficients, and divide by D:
To find y substitute constants into y co-efficients and divide by D:
See, it's as simple as that! Just for good measure, I'll do an example using 3 unknowns in 3 equations:
Solve for x:
Solve for y:
Solve for z:
And there we have it, how to solve a series of simultaneous equations using determinants, something that can be very useful.
Matrix Inversion
Equations can also be solved by inverting a matrix. Take the following equations again:
We push these into 3 matrices to solve:
Let's give these names such that:
We need to solve for B, and since there is no "matrix divide" operation, we need to invert the matrix A and multiply it by D, such that:
Now we need to know how to actually do the matrix inversion. There are many ways to do this, and the way I am going to show you. Let's find the first element in a 3x3 matrix. Let's call it c11. We need to get rid of the row and column that intersects this, so that:
c11 will then take the value of the following determinant:
The sign in front of c11 is decided by the expression:
Where i and j are the positions of the element in the matrix.
That's easy enough isn't it? Thought so J. Just do the same for every element, and build up the co-factor matrix. Done? Good. Now that the co-factor matrix has been found, the inverse matrix can be calculated using the following equation:
Taking the previous example and equations, let's find the inverse matrix of A.
First, the co-factor matrix C would be:
and |A| is:
So A-1 is:
To solve the equations, we then do:
We can then find the values of x,y and z by pulling them out of the last matrix, such that x = -62, y = 39 and z = 3, which is what the other method using determinants found.
A matrix is called orthogonal if its inverse equals its transpose.
Matrices in computer graphics
All graphics APIs use a set of matrices to define transformations in space. A transformation is a change, be it translation, rotation, or whatever. Using column a matrix to define a point in space, a vertex, we can define matrices that alter that point in some way.
Transformation Matrices
Most graphics APIs use 3 different types of primary transformations. These are:
Translation
Scale
Rotation
I won't go into the derivation of the matrices for these transformations, as that will take up far too much space. Any good math book that explains affine space transformations will explain their derivations. You have to pre-multiply points by the transformation matrix, as it is impossible to post-multiply because of the dimensions.
Therefore, a point p can be transformed to point p' using a transformation matrix T so that:
Translation
To translate a point onto another point, there needs to be a vector of movement, so that
where p' is the translated point, p is the original point, and v is the vector along which to translate.
In matrix form, this turns into:
Where dx, dy and dz are the components of the vector in the respective axis of movement. Note that a 4D vertex is used. These are called homogeneous co-ordinates, BUT I will not discuss them here.
Scaling
You can scale a vertex by multiplying it by a scalar value, so that
where k is the scalar constant. You can multiply each component of p by a different constant. This will make it so you can scale each axis by a different amount.
In matrix form this is:
Rotation
Rotation is the most complex transformation. Rotation can be performed around the 3 Cartesian axes.
The rotation matrices around these axis are:
To find out more about how these matrices are derived, please pick up a good math book, I haven't got the time to write it here. Some things about these matrices though:
Any rotation about an axis by q can be undone by a successive rotation by -q. So:
Also, notice that the cosine terms are always on the top-left to bottom-right diagonal, and the sine terms are always on the top-right to bottom-left diagonal, we can also say that:
Rotations matrices that act about the origin are orthogonal.
Note that these transformations are cumulative. That is, if you multiplied a vertex by a translation matrix, then by a scale matrix, it would have the effect of moving the vertex, then scaling it. The order that you multiply becomes very important when you multiply rotation and translation matrices together, as RT does NOT equal TR!
Projection Matrices
These are also complicated matrices. They come in two flavours, perspective correct and orthographic. There are some very good books that derive these matrices in an understandable way, so I won't cover it here. Since I don't work with projection matrices very often, I had to look a lot of this material up using the book Interactive Computer Graphics by Edward Angel. A very good book that I suggest you buy.
Anyway, on to the matrices.
The orthographic projection matrix:
The x, y and zmax/min variables define the viewing volume.
The perspective correct projection matrix is:
Conclusion
Well, that's it for this tutorial. I hope that I've helped you understand vectors and matrices including how to use them. For further reading I can recommend a few books that I have found really useful, these are:
Interactive Computer Graphics – A Top Down Approach with OpenGL" – Edward Angel: Covers a lot of theory in computer graphics, including how we perceive the world around us. This book covers a lot of the matrix derivations that I left out. All in all, a very good book on graphics programming and theory. With exercises too, which is nice.
Mathematics for Computer Graphics Applications – Second Edition – M.E Mortenson: This is solely about the mathematics behind computer graphics, and explains a lot of material in a very easy to understand manner. There are loads of exercises to keep you occupied. The book explains things such as vectors, matrices, transformations, topology and continuity, symmetry, polyhedra, half-spaces, constructive solid geometry, points, lines, curves, surfaces, and more! A must for anyone serious in graphics programming. You won't see a line of code or pseudo-code though.
Advanced National Certificate Mathematics Vol.: 2 – Pedoe: I don't know whether you can actually get this book anymore, but if you can get a copy! This book explains mathematical concepts well, and is easy to learn from. This book is about general mathematics though, each volume expands on the other. So vol. 1 introduces concepts, vol. 2 expands on them. A book well worth the money (although I have no idea how much it is, as I got my copy off my dad ).
That's about it! I hope I haven't scared you off graphics programming. Most APIs, including Direct3D and OpenGL, will hide some of this away from you.
If you need to contact me at all, my email address is: phil.dadd@btinternet.com. I don't want any abuse though - if you don't like this tutorial I accept constructive advice only.
Credits
I'd like to give credit to "Advanced National Certificate Mathematics Vol.: 2" as that's where I got the simultaneous equations from in the part on determinants, so I knew the answers were whole, and that they worked out. I would also like to give credit to Miss. A Miller who proof read this tutorial for me.]]>Tue, 04 Jun 2002 23:47:26 +0000d20d9896e5f7a733d09c07acb323154fThe Matrix and Quaternions FAQ
(Editor's Note - This article requires additional formatting work and is incomplete)
Version 1.21 30th November 2003 ------------------------------- Please mail feedback to matrix_faq@j3d.org with a subject starting with MATRIX-FAQ
(otherwise my spam filter will simply kill your message). Any additional suggestions or related questions are welcome. Just send E-mail to the above address. The latest copy of this FAQ can be found at the following web page:
History ------- I (Andreas) tried to find "hexapod@(no-spam)netcom.com" who seemed to have maintained this for a while, but the site at netcom.com doesn't exist anymore, emails bounce. Since I (and colleques) wasted quite some time figuring out what was wrong with some of the algorithms given in the earlier versions of this document, I decided to correct it and put it back on the web. The formerly given sites for the location of these documents do not exist anymore: ftp://ftp.netcom.com/pub/he/hexapod/index.html Versions, dates and links to local copies (so you can compare): matrfaq_1.02.html: Version 1.2 2nd September 1997 matrfaq_1.04.html: Version 1.4 26th December 1998 matrfaq_1.06.html: Version 1.6 30th September 2000 matrfaq_1.07.html: Version 1.7 20th December 2000 matrfaq_1.08.html: Version 1.8 21th December 2000 matrfaq_1.09.html: Version 1.9 16th January 2001 matrfaq_1.10.html: Version 1.10 30th January 2001
matrfaq_1.11.html: Version 1.11 9th February 2001 matrfaq_1.12.html: Version 1.12 26th March 2001 matrfaq_1.13.html: Version 1.13 20th July 2001 matrfaq_1.14.html: Version 1.14 17th August 2001 matrfaq_1.15.html: Version 1.15 20th August 2001 matrfaq_1.16.html: Version 1.16 2nd October 2001 matrfaq_1.17.html: Version 1.17 30th November 2001 matrfaq_1.18.html: Version 1.18 27th January 2002 matrfaq_1.19.html: Version 1.19 20th March 2002
matrfaq_1.20.html: Version 1.20 31st January 2002 matrfaq_1.21.html: Version 1.21 30th November 2003
Please refrain from asking me math questions. I am only maintaining this FAQ and have very little knowledge about the subject. But, if you have a question that is not answered by this FAQ and later happen to find the answer and believe it to be relevant for this FAQ (or its readers), please send all relevant information, hopefully in a pre-digested form, to me to be included here. Thanks! If you prefer to appear as "anonymous" in the contributions list, let me know, otherwise I'll just put you down with whatever name I can gather from your email header.
Q32. What are yaw, roll and pitch? Q33. How do I combine rotation matrices? Q34. What is Gimbal Lock? Q35. What is the correct way to combine rotation matrices? Q36. How do I generate a rotation matrix from Euler angles? Q37. How do I generate Euler angles from a rotation matrix? Q38. How do I generate a rotation matrix for a selected axis and angle? Q39. How do I generate a rotation matrix to map one vector onto another? Q40. How do I use matrices to convert between two coordinate systems?
Q41. What is a translation matrix? Q42. What is a scaling matrix? Q43. What is a shearing matrix? Q44. How do I perform linear interpolation between two matrices? Q45. How do I perform cubic interpolation between four matrices? Q46. How can I render a matrix?
QUATERNIONS =========== Q47. What are quaternions? Q48. How do quaternions relate to 3D animation?
Q49. How do I calculate the conjugate of a quaternion? Q50. How do I calculate the inverse of a quaternion? Q51. How do I calculate the magnitude of a quaternion? Q52. How do I normalise a quaternion? Q53. How do I multiply two quaternions together? Q54. How do I convert a quaternion to a rotation matrix? Q55. How do I convert a rotation matrix to a quaternion? Q56. How do I convert a rotation axis and angle to a quaternion? Q57. How do I convert a quaternion to a rotation axis and angle?
Q58. How do I convert spherical rotation angles to a quaternion? Q59. How do I convert a quaternion to spherical rotation angles? Q60. How do I convert Euler rotation angles to a quaternion? Q61. How do I use quaternions to perform linear interpolation between matrices? Q62. How do I use quaternions to perform cubic interpolation between matrices? Q63. How do I use quaternions to rotate a vector?
Introduction ------------
I1. Important note relating to OpenGl and this document ------------------------------------------------------- In this document (as in most math textbooks), all matrices are drawn in the standard mathematical manner. Unfortunately graphics libraries like IrisGL, OpenGL and SGI's Performer all represent them with the rows and columns swapped. Hence, in this document you will see (for example) a 4x4 Translation matrix represented as follows:
OpenGL uses a one-dimensional array to store matrices - but fortunately, the packing order results in the same layout of bytes in memory - so taking the address of a pfMatrix and casting it to a float* will allow you to pass it directly into routines like glLoadMatrixf. In the code snippets scattered throughout this document, a one-dimensional array is used to store a matrix. The ordering of the array elements is transposed with respect to OpenGL.
I2. Important note with respect to normalized inputs ---------------------------------------------------- Note that most algorithms assume normalized inputs, such as vectors of union length, or matrices with normalized main diagonal etc. It is possible, and often enough the case, that algorithms (and the code snippets provided here) work correctly with arbitrary inputs, but it is usually considered bad practise (and you will pay in debugging time if you fail to observe this suggestion) to rely on this property.
Answers -------
BASICS ======
Q1. What is a matrix? ---------------------- A matrix is a two dimensional array of numeric data, where each row or column consists of one or more numeric values. Arithmetic operations which can be performed with matrices include addition, subtraction, multiplication and division. The size of a matrix is defined in terms of the number of rows and columns. A matrix with M rows and N columns is defined as a MxN matrix. Individual elements of the matrix are referenced using two index values. Using mathematical notation these are usually assigned the variables 'i' and 'j'. The order is row first, column second For example, if a matrix M with order 4x4 exists, then the elements of the matrix are indexed by the following row:column pairs:
| 00 01 02 03 | M = | 10 11 12 13 | | 20 21 22 23 | | 30 31 32 33 |
The element at the top right of the matrix has i=0 and j=3 This is referenced as follows:
M = M i,j 0,3
In computer animation, the most commonly used matrices have either 2, 3 or 4 rows and columns. These are referred to as 2x2, 3x3 and 4x4 matrices respectively. 2x2 matrices are used to perform rotations, shears and other types of image processing. General purpose NxN matrices can be used to perform image processing functions such as convolution. 3x3 matrices are used to perform low-budget 3D animation. Operations such as rotation and multiplication can be performed using matrix operations, but perspective depth projection is performed using standard optimised into pure divide operations. 4x4 matrices are used to perform high-end 3D animation. Operations such as multiplication and perspective depth projection can be performed using matrix mathematics.
Q2. What is the "order" of a matrix? ------------------------------------- The "order" of a matrix is another name for the size of the matrix. A matrix with M rows and N columns is said to have order MxN.
Q3. How do I represent a matrix using the C/C++ programming languages? ----------------------------------------------------------------------- The simplest way of defining a matrix using the C/C++ programming languages is to make use of the "typedef" keyword. Both 3x3 and 4x4 matrices may be defined in this way ie:
typedef float MATRIX3[9]; typedef float MATRIX4[16];
Since each type of matrix has dimensions 3x3 and 4x4, this requires 9 and 16 data elements respectively. At first glance, the use of a single linear array of data values may seem counter-intuitive. The use of two dimensional arrays may seem more convenient ie.
typedef float MATRIX3[3][3]; typedef float MATRIX4[4][4];
However, the use of two reference systems for each matrix element very often leads to confusion. With mathemetics, the order is row first (i), column second (j) ie.
Mij
Using C/C++, this becomes
matrix[j][i]
Using two dimensional arrays also incurs a CPU performance penalty in that C compilers will often make use of multiplication operations to resolve array index operations. So, it is more efficient to stick with linear arrays. However, one issue still remains to be resolved. How is an two dimensional matrix mapped onto a linear array? Since there are only two methods (row first/column second or column first/row column). The performance differences between the two are subtle. If all for-next loops are unravelled, then there is very little difference in the performance for operations such as matrix-matrix multiplication. Using the C/C++ programming languages the linear ordering of each matrix is as follows:
Q6.[nbsp][nbsp]What is the identity matrix? --------------------------------- [nbsp][nbsp]The identity matrix is matrix in which has an identical number of rows [nbsp][nbsp]and columns. Also, all the elements in which i=j are set one. All others [nbsp][nbsp]are set to zero. For example a 4x4 identity matrix is as follows:
Q7.[nbsp][nbsp]What is the major diagonal of a matrix? -------------------------------------------- [nbsp][nbsp]The major diagonal of a matrix is the set of elements where the [nbsp][nbsp]row number is equal to the column number ie.
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Product Description
Review
R. Hartshorne Algebraic Geometry "Enables the reader to make the drastic transition between the basic, intuitive questions about affine and projective varieties with which the subject begins, and the elaborate general methodology of schemes and cohomology employed currently to answer these questions."—MATHEMATICAL REVIEWS
This book is one of the most used in graduate courses in algebraic geometry and one that causes most beginning students the most trouble. But it is a subject that is now a "must-learn" for those interested in its many applications, such as cryptography, coding theory, physics, computer graphics, and engineering. That algebraic geometry has so many applications is quite amazing, since it was not too long ago that it was thought of as a highly abstract, esoteric topic. That being said, most of the books on the subject, including this one, are written from a very formal point of view. Those interested in applications will have to face up to this when attempting to learn the subject. To read this book productively one should gain a thorough knowledge of commutative algebra, a good start being Eisenbud's book on this subject. Also, it is important to dig into the original literature on algebraic geometry, with the goal of gaining insight into the constructions and problems involved. The author of this book does not make an attempt to motivate the subject with historical examples, and so such a perusal of the literature is mandatory for a deeper appreciation of algebraic geometry. The study of algebraic geometry is well worth the time however, since it is one that is marked by brilliant developments, and one that will no doubt find even more applications in this century.
Varieties, both affine and projective, are introduced in chapter 1. The discussion is purely formal, with the examples given unfortunately in the exercises. The Zariski topology is introduced by first defining algebraic sets, which are zero sets of collections of polynomials. The algebraic sets are closed under intersection and under finite unions. Therefore their complements form a topology which is the Zariski topology. The properties of varieties are discussed, along with morphisms between them. "Functionals" on varieties, called regular functions in algebraic geometry, are introduced to define these morphisms. Rational and birational maps, so important in "classical" algebraic geometry are introduced here also. Blowing up is discussed as an example of a birational map. A very interesting way, due to Zariski, of defining a nonsingular variety intrinsically in terms of local rings is given. The more specialized case of nonsingular curves is treated, and the reader gets a small taste of elliptic curves in the exercises. A very condensed treatment of intersection theory in projective space is given. The discussion is primarily from an algebraic point of view. It would have been nice if the author would have given more motivation of why graded modules are necessary in the definition of intersection multiplicity.
The theory of schemes follows in chapter 2, and to that end sheaf theory is developed very quickly and with no motivation (such as could be obtained from a discussion of analytic continuation in complex analysis). Needless to say scheme theory is very abstract and requires much dedication on the reader's part to gain an in-depth understanding. I have found the best way to learn this material is via many examples: try to experiment and invent some of your own. The author's discussion on divisors in this chapter is fairly concrete however.
The reader is introduced to the cohomology of sheaves in chapter 3, and the reader should review a book on homological algebra before taking on this chapter. Derived functors are used to construct sheaf cohomology which is then applied to a Noetherian affine scheme, and shown to be the same as the Cech cohomology for Noetherian separated schemes. A very detailed discussion is given of the Serre duality theorem.
Things get much more concrete in the next chapter on curves. After a short proof o the Riemann-Roch theorem, the author studies morphisms of curves via Hurwitz's theorem. The author then treats embeddings in projective space, and shows that any curve can be embedded in P(3), and that any curve can be mapped birationally into P(2) if one allows nodes as singularities in the image. And then the author treats the most fascinating objects in all of mathematics: elliptic curves. Although short, the author does a fine job of introducing most important results.
This is followed in the next chapter by a discussion of algebraic surfaces in the last chapter of the book. The treatment is again much more concrete than the earlier chapters of the book, and the author details modern formulations of classical constructions in algebraic geometry. Ruled surfaces, and nonsingular cubic surfaces in P(3) are discussed, as well as intersection theory. A short overview of the classification of surfaces is given. The reader interested in more of the details of algebraic surfaces should consult some of the early works on the subject, particularly ones dealing with Riemann surfaces. It was the study of algebraic functions of one variable that led to the introduction of Riemann surfaces, and the later to a consideration of algebraic functions of two variables. A perusal of the works of some of the Italian geometers could also be of benefit as it will give a greater appreciation of the methods of modern algebraic geometry to put their results on a rigorous foundation.
33 of 36 people found the following review helpful
5.0 out of 5 starsTHE book for the Grothendieck approach16 Mar 2004
By Davis C. Doherty - Published on Amazon.com
Format:Hardcover
This is THE book to use if you're interested in learning algebraic geometry via the language of schemes. Certainly, this is a difficult book; even more so because many important results are left as exercises. But reading through this book and completing all the exercises will give you most of the background you need to get into the cutting edge of AG. This is exactly how my advisor prepares his students, and how his advisor prepared him, and it seems to work.
Some helpful suggestions from my experience with this book: 1) if you want more concrete examples of schemes, take a look at Eisenbud and Harris, The Geometry of Schemes; 2) if you prefer a more analytic approach (via Riemann surfaces), Griffiths and Harris is worth checking out, though it lacks exercises.
43 of 49 people found the following review helpful
5.0 out of 5 starsTerrific, if you want it.24 Sep 2000
By Colin McLarty - Published on Amazon.com
Format:Hardcover
This book hardly needs a review on Amazon, because if you have as much math background as it needs, then you must already know it is indispensible for learning about schemes in algebraic geometry. The book is clear, concise, very well organized, and very long. If you do not already know the Noether normalization theorem, and the Hilbert Nullstellensatz, then you do not want this book yet--you want an introduction to commutative algebra.
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Peer Review
Ratings
Overall Rating:
This webpage has all lecture material, homework, homework solutions, and computer laboratories for Calculus that would be needed for a Biology course, so it should be self contained.
Learning Goals:
The emphasis is on mathematical modeling of biological systems.
Target Student Population:
Calculus I students (biology...but very pertinent to faculty who should see how to show students that calculus is relevant and interesting.)
Prerequisite Knowledge or Skills:
Understanding of precalculus
Type of Material:
Lecture/Presentation
Recommended Uses:
Teachers and Calculus I students.
Technical Requirements:
Level four or higher browser. To do the homework the following is required: spreadsheet, wordprocessor, some exercises require more poserful software.
Evaluation and Observation
Content Quality
Rating:
Strengths:
Excellent! This is a very comprehensive site. The Cricket Thermometer (listening to crickets on the web, then using a linear model for relating to temperature) is one of several interesting problems. Dr. Mahaffy's laboratory experiences are well designed; pre- and in-service teachers find them very interesting. The reviewers view the value of this web site as an excellent model for teachers of mathematics to apply regardless of the mathematical topic that is presented.
Concerns:
Prehaps teachers will not take the time to investigate the episodes on the web site, because they will view the content as too advanced for their use. The reality is that the applications to lessons for grades 4-20 are bountiful.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
The Internet-based course is designed to show how calculus naturally arises in biological examples from classical and current research. The graphics are excellent for supporting understanding and the animations provide a semi-concrete experience for both pre-service and in-service teachers and calculus students. Dr. Mahaffy's web-based course uses convincing examples and has a significant portion where technology can aid in teaching more complicated models. Also, this site is good for expanding teacher's ideas of how to present the lessons to their own students.
Concerns:
Teachers will need to develop their own rubrics for evaluating the products completed for the laboratories/homework. Again, this is not a concern, but the user might preceive it to be of less value, because the rublics are not included.
This site should not be viewed as a stand-alone learning tool for classes. (It may be that individuals use it for independent study,
but the users' experiences need to involve group work, teacher instruction, and individual study as they study calculus for biology.
Users must be actively involved. These are not rote skills that are being presented. They are mathematical/scientific concepts. If "participants" merely watch the animations, they will not be helped much toward understanding the concept/principle being presented. If (future) teachers take the next step and do the exercises, the teacher's own understanding should be greatly facilitated.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
Requires the use of a level 4(or higher) Java-enabled browser.
Most of the mathematics/biology homework will be done with the spreadsheet software, Excel, and wordprocessor, Word.
High school mathematics teachers used one of the episodes on Dr. Mahaffey's Calculus for Biology web site. They enjoyed it. Their Internet experience was supplemented with a group discussion led by a mathematics professor.
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mathematics can be a challenge for anyone. Math Flight can help you master it with three fun activities to choose from! With lots of graphics and sound effects, your interest in learning math should never decline.
GrafEq (pronounced 'graphic') is an intuitive, flexible, precise and robust program for producing graphs of implicit relations. GrafEq is designed to foster a strong visual understanding of mathematics by providing reliable graphing technology.
GraphiCal is a programmable graphics calculator which lets you visualize expressions and formulas as graphs in a chart. Creates animated video clips from a sequence of graphs. Built-in functions (>50) include integration, root finding ..
This is an advanced expression and conversion calculator. Vast array of built-in functions, constants and confersion operations that can be extended with your own user-defined functions. Now with graphs.
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actuary-- assemble and analyze statistics to calculate probabilities of death, sickness, injury, disability, unemployment, retirement, and property loss; design insurance and pension plans and ensure that they are maintained on a sound financial basis
mathematics teacher-- introduce students to the power and beauty of mathematics in elementary, junior high, or high school mathematics courses
physician-- diagnose patient illnesses, prescribe medication, teach classes, mentor interns, and do clinical research; students with a good mathematics background will find themselves being admitted to the best medical schools and discover that mathematics has prepared them well for the discipline, analysis, and problem- solving required in the field of medicine
research scientist-- model atmospheric conditions to gain insight into the effect of changing emissions from cars, trucks, power plants, and factories; apply these models in the development of alternative fuels
staff systems air traffic control analyst-- apply probability, statistics, and logistsics to air traffic control operations; use simulated aircraft flight to monitor air traffic control computer systems
cryptologist-- design and analyze schemes used to transmit secret information
attorney-- research, comprehend, and apply local, state, and federal laws; a good background in mathematics will help a student get admitted to law school and assist in the understanding of complicated theoretical legal concepts
economist-- interpret and analyze the interrelationships among factors which drive the economics of a particular organization, industry, or country
mathematics professor-- teach mathematics classes, do theoretical research, and advise undergraduate and graduate students at colleges and universities
environmental mathematician-- work as member of interdisciplinary team of scientists and professionals studying problems at specific Superfund sites; communicate effectively across many academic discilplines and be able to summarize work in writing
robotics engineer-- combine mathematics, engineering, and computer science in the study and design of robots
geophysical mathematician -- develop the mathematical basis for seismic imaging tools used in the exploration and production of oil and gas reservoirs
design -- use computer graphics and mathematical modeling in the design and construction of physical prototypes; integrate geometric design with cost-effective manufacturing of resulting products
ecologist -- study the interrelationships of organisms and their environments and the underlying mathematical dynamics
geodesist -- study applied science involving the precise measurement of the size and shape of the earth and its gravity field (courtesy of Bruce Hedquist)
photogrammetrist -- study the applied science of multi-spectral image acquisition from terrestrial, aerial and satellite camera platforms, followed up by the image processing, analysis, storage, display, and distribution in various hard-copy and digital format (courtesy of Bruce Hedquist)
civil engineer -- plan, design, and manage the construction of land vehicle, aircraft, water, and energy transport systems; analyze and control systems for land vehicular traffic; analyze and control environmental systems for sewage and water treatment; develop sites for industrial, commercial and residential home use; analyze and control systems for storm water drainage and storage; manage construction of foundations, structures and buildings; analyze construction materials ; and surface soils and subterranean material analysis (courtesy of Bruce Hedquist)
geomatics engineer -- once known as "surveying engineer", includes geodetic surveying : takes into account the size and shape of the earth, in order to determine the precise horizontal and vertical positions of geodetic reference monuments; cadastral surveying : establishes and reestablishes the reference monuments for the U.S. Public Land Survey System, i.e., township and section corners; topographic surveying : determines the detailed configuration or contour of the natural earth's surface and the position of fixed objects thereon or related thereto; hydrographic surveying : similarly determines underwater contours and features; land surveying : is the location of existing parcel and new land subdivision lines, road and utility rights-of-way and easement lines, and determination of the location of existing and new reference monuments, which mark property lines and parcel corners; land surveying : also involves the preparation of legal descriptions for officially recorded land ownership conveyance deeds and other land title documents; construction surveying : is the determination of the direction and length between and the elevations of reference points for fixed private and public works, as embraced within the definition and practice of civil engineering, and the labeling of reference markers containing critical information for the construction thereof; design, operation and management of advanced Geographic Information Systems (GIS and Land Information Systems (LIS), as well as other sophisticated computer mapping and CAD based geospatial applications (courtesy of Bruce Hedquist)
EDIT: Well, as the person said in below, you will more than likely need a degree for the actual position of most of the positions listed above. As I said in the beginning, those are fields mathematical knowledge WILL COME IN HANDY. If you just degreed in mathematics, it won't mean you will have the sufficient knowledge to do the jobs such as computer programmer. You will also need to know how the knowledge of the computer language(s) to successfully do the job but knowing math and having better understanding of it will give you an edge. That is pretty much common sense.
Answered By: Jason - 2/22/2007
Additional Answers
()
Accounting
civil engineer
Answered By: sassy_czar - 2/22/2007
I hesitate to respond, because ... ever hear the saying "on the Internet, nobody knows that you are a dog"? One will write truthfully about the difficulties in a field, and some trollish contrarian will show up and talk at length about how he hasn't witnessed any of those difficulties during his glorious career. Fairly often, on prolonged questioning, our corporate hero can't maintain the illusion that he has much of a background in the field he claims to have a career in, but Internet exchanges tend to be brief, so online, the snow job works. In an oh-so-postmodern way, the group will say something like "you have your experiences and he has his, so why don't you accept that both are legitimate", not really trying that hard to understand the exasperation of somebody who is finding that an honest telling of the unhappy truth is being put on an equal level with a shameless fabrication. Why some people enjoy doing this, I'm not entirely sure, but all the same, and probably against my better judgment, I'll give you an honest answer, even though it probably is not one you're going to want to hear or accept.
What are the job opportunities available in the field of Mathematics? That largely depends on location, probably. The American job market is the one I'm familiar with, so I'll focus on that one in my reply. Your name sounds Indian, but then again, so do those of a great many of my neighbors in Chicago, so I could only guess as to where you live, possibly doing so with no great success.
If you were here and asking me that question, my response would be "do not major in Mathematics, and certainly don't go to graduate school in the subject". The problem is that while an academic background in this field opens few doors, if any, when it comes to employment, it closes a good many forever. One is refused what employment is available on the basis of "overqualification", discovering that the hard work one did in pursuing an education as one worked one's way to school has been rewarded with the equivalent of a blacklisting, while those who partied their way through frat party after frat party go sailing into high paying positions based on the personal connections they made. Which sounds like a better deal?
This is not a temporary inconvenience. Imagine looking for work, not for days or weeks or months or even for years, but for decades in a system that, while deeply concerned with making certain that convicted felons can find work, sees absolutely nothing wrong with leaving you to starve or freeze to death, not in spite of the fact that you worked harder and learned more than your peers in a particular subject, but because of this fact.
Looking at some of the suggestions I've seen in this thread, some people are just blowing smoke. Accounting is an entirely distinct degree program. Knowing how to solve a PDE, for example, is not going to teach you anything about how to help a client make best use of the tax laws. Civil engineers need to have civil engineering degrees, and rightly so.
Software engineering - One of my brothers, degreed in the relevant field (computer science) with extensive experience has been doing better than I have with my on again - off again tutoring career, but at this point he works long hours at minimum wage. Data entry was the only work he could find; there are almost no software engineering jobs to be found in our area (Chicago).
Yes, there are want ads, but in the real world most of these seem to be for positions that on examination, don't really exist, or call for a long list of narrow qualifications that would eliminate almost every candidate in existence. No less than 10 years of experience and no more than 12 programming in a language that was only invented 5 years ago, that sort of nonsense.
Besides which, Mathematics is not the degree for that profession, Computer Science is.
Actuarial work - A little more understandable as a suggestion, as these people are doing statistics for a living, but not really much more feasible. The insurance industry was one of the ones that went on a kick of refusing to hire anybody without 2-5 years of relevant job experience, remaing quite vague about how it was, precisely, that newcomers to the field were supposed to get that experience, if they were never going to be allowed the chance to get their first job.
About half of a decade later, one could hear placement people in that field whining about the shortage of junior actuarial personnel, as if there were something deeply mysterious about the phenomenon of running out of new people in a field, when one has spent years stubbornly refusing to let new people have a chance to enter the field. It's sort of like watching somebody refuse to go to the store, and then scream about the unfairness of life when he discovers that there is no food in his refrigerator.
A lucky, well connected few manage to bypass this insanity, but the thing about good luck is that it is, by definition, scarce.
"mathematician" - and good luck finding a position under that title, and then beating out the stampede of long term unemployed and underemployed mathematicians looking for work.
"mathematics teacher"-- School districts are downsizing, and one needs a degree in Education for that, not a degree in Mathematics.
"physician"-- Somebody is dreaming. Medicine is based in Biology, not Mathematics.
"mathematics professor"-- Welcome to adjunct Hell. Full-time, tenure track employment for new people is largely a thing of the past.
etc. etc., etc. Most of the professions I saw listed would require degrees in fields other than Mathematics, a number are notorious in the real world for downsizing that has been going on for years or decades, and some of which seem to be nothing more than flights of fancy on the part of a very creative author who imagines that such a position MUST exist, and sees no harm in publishing his guesswork as fact. Environmental Mathematician? Try finding a SPECIFIC position offered under that title.
I'm guessing that I'm seeing excerpts from some kind of occupational handbook, but the disgraceful reality about those handbooks is that the authors tend to gut things out without doing any fieldwork of their own, and without having any knowledge of the occupations they write about, with such an air of authority. I guess this must offer the writers an easy living, one that won't trouble them too much, as Professionalism seems to be widely considered to be a quaint relic of the pre-modernist past, clung to by those who aren't wise enough to understand that perception is reality and that the Market is God, and so still draw a distinction between earning an honorable livelihood and earning a dishonorable one. This is the dark side of "do your own thing" - those who one should be able to trust, proving so utterly untrustworthy and never being called to task for this.
Such is reality, and Heaven help you if you fail to act on it swiftly, because your fellow man never will. Get out of this field while you still have a chance.
Other Career Questions
I'm from California, but I'm currently in Hawaii and some people who teach here are saying that in the USA those jobs are thinning out. They say our leaders are hiding problems, like reasons for the NASA cuts. I know CA is having lots of problems, but I'm not too familiar with what goes on in other states. I looked online and the prospects seem good right now. Will it continue to be this way, or are they really thinning out?
I am trying to apply to the business school at my college but if i dont get in i will have to minor in business instead. if i major in mathematics what jobs can i get with that and a minor in accounting? Do these jobs pay well?
I am good at mathematics and am trying to find out a fun job that pays $100 k plus involving mathematics. Are you working this type of job? If so how did you get started? Do you know someone with this type of job? What is their job like? If you dont know please dont programs
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Middle School Math 2 continues to build on the concepts introduced in seventh grade. Students will continue to deepen their understanding of mathematics in preparation for high school mathematics. Students will continue to explore and solve mathematical problems, think critically, work cooperatively with others, and communicate their ideas clearly as they work through mathematical concepts. A summary of the major concepts and procedures learned in this course follows. Students will work with lines and angles, especially as they solve problems involving triangles, using square roots and the Pythagorean Theorem. In eighth grade, students will solve a variety of linear equations and inequalities. They will represent and determine the slope and y-intercept of linear functions with verbal descriptions, tables, graphs and symbolic expressions. Students will work with lines and angles, especially as they solve problems involving triangles, using square roots and the Pythagorean Theorem. Students will build on their extensive experience organizing and interpreting data, by using mean, median, and mode to analyze, summarize, and describe information. Additionally, students will be introduced to scientific notation, the laws of exponents, and irrational numbers
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Graphing Calculator Manual For Stats - 3rd edition
Summary: Organized to follow the sequence of topics in the text, this manual is an easy-to-follow, step-by-step guide on how to use the TI-83/84 Plus and TI-89 graphing calculators. It provides worked-out examples to help students fully understand and use their graphing calculator.
Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book
$1617.71 +$3.99 s/h
VeryGood
AlphaBookWorks Alpharetta, GA
0321570944
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