Mathematics for game developers

649 Pages·2004·10.76 MB·English

by  Christopher Tremblay

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To the only things that truly matter:
family and friends
Acknowledgments
So many people were involved in bringing this book to life, thanking each person
individually would be a book in itself.Instead ofsubjecting you to that,I’ll break the
deserving into groups. From the early-on mathematicians to the old-school demo
programmer gurus who taught me the secrets ofthe PC,I thank you all for the invaluable
technical knowledge. Special thanks to the University of Ottawa professors who could
actually teach,a rarity in an educator (especially in the engineering field).Thanks to my
personal stress relievers (that is,the Quake3knobs on XO server),which were like caffeine
when the hours were late.Special thanks to everyone who shepherded this book through
production;I think the end result speaks for itself.Finally,I would like to thank every sin-
gle reader out there,without whom none ofthis would be possible.After you’ve mastered
the material here,I call on you to create some kick-ass demo to make us all proud!
iv
About the Author
CHRISTOPHER TREMBLAY lives in the California Bay Area, where he works for Motorola
building a 3D graphics engine for cell phones to power next-generation games.He holds
a degree in Software Engineering from the University ofOttawa,Canada,and is currently
completing final courses for a mathematics degree. His work in the game industry
includes game AI, core-networking, software rendering algorithms, 3D geometry algo-
rithms,and optimization.Although most ofhis work is PC-based,a fair amount ofit was
done on embedded devices ranging from bottom-line TI-calculators to Z80 and 68K Palm
processors to speedy PocketPC strong-arm processors,with games such as LemmingZ.
About the Series Editor
ANDRÉLAMOTHE,CEO,Xtreme Games LLC,has been involved in the computing indus-
try for more than 25 years.He wrote his first game for the TRS-80 and has been hooked
ever since! His experience includes 2D/3D graphics, AI research at NASA, compiler
design,robotics,virtual reality,and telecommunications.His books are top sellers in the
game programming genre,and his experience is echoed in the Thomson Course Technol-
ogy PTR Game Developmentseries.
v
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Part I: The Basics 1
Chapter 1 Equation Manipulation and Representation. . . . . . . . . . . . . 3
Choosing a Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4
Equation Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
Using Polynomial Roots to Solve Equations . . . . . . . . . . . . . . . . . . . .18
Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
Chapter 2 Baby Steps: Introduction to Vectors . . . . . . . . . . . . . . . . . . 29
O Vector, What Art Thou? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29
Basic Operations and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .31
Advanced Operations and Properties . . . . . . . . . . . . . . . . . . . . . . . .36
Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
Chapter 3 Meet the Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Meat, the Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .54
Basic Operations and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
Advanced Operations and Properties . . . . . . . . . . . . . . . . . . . . . . . .64
Matrix Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .81
Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .89
vi
Contents vii
Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .91
Vector Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92
Chapter 4 Basic Geometric Elements. . . . . . . . . . . . . . . . . . . . . . . . . . 95
Creating Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96
Generating Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98
Constructing Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101
When Elements Collide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103
Know Your Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115
3D File Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .123
Chapter 5 Transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
It’s All About the Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126
Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128
Multiple Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . .138
Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .144
Non-Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151
Chapter 6 Moving to Hyperspace Vectors: Quaternions. . . . . . . . . . 153
Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .154
Basic Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .158
Advanced Operations on Quaternions . . . . . . . . . . . . . . . . . . . . . . .161
Part II: Physics 167
Chapter 7 Accelerated Vector Calculus for the Uninitiated. . . . . . . . 169
The Concept of Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .170
Derivatives (The Result of Differentiation) . . . . . . . . . . . . . . . . . . .175
Reversing: Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191
Chapter 8 Gravitating Around Basic Physics . . . . . . . . . . . . . . . . . . . 209
Move That Body... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .210
Physical Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .215
Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .227
Chapter 9 Advanced Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Angular Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .236
viii Contents
Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .242
Center of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .257
Angular Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .259
Chapter 10 And Then It Hits You: You Need Collision Detection . . . . 263
Environmental Collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .264
Collisions of Objects Among Themselves . . . . . . . . . . . . . . . . . . . . .269
Picking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287
Part III: Playing with Numbers 293
Chapter 11 Educated Guessing with Statistics and Probability. . . . . . 295
Basic Statistics Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .296
Random Number Generation (Uniform Deviates) . . . . . . . . . . . . . .302
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .310
Intelligence Everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .319
Chapter 12 Closing the Gap for Numerical Approximation. . . . . . . . . 323
Solution Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324
2D Function Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .327
Multivariable Function Approximation . . . . . . . . . . . . . . . . . . . . . .358
Chapter 13 Exploring Curvy Bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363
Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .385
Part IV: Rendering and Lighting 389
Chapter 14 Graphics-Generation Engines . . . . . . . . . . . . . . . . . . . . . . 391
Decals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .392
Billboards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .397
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .402
Chapter 15 Visibility Determination: The World of the Invisible . . . . 405
GPU-Level Visibility Determination . . . . . . . . . . . . . . . . . . . . . . . . .406
CPU Culling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .409
Grid Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .427
Contents ix
Chapter 16 Space Partitioning: Cleaning Your Room . . . . . . . . . . . . . 427
Quadtrees and Octrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .430
k-D Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .431
Binary Space Partitioned (BSP) Trees . . . . . . . . . . . . . . . . . . . . . . . .437
Portal Subdivision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .441
Chapter 17 Filling the Gaps with Rendering Techniques. . . . . . . . . . . 447
Studying the 3D Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .448
Texture Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .450
Cube Map Tricks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .463
Cel Shading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .466
Shadows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .468
Chapter 18 Building a Light Show. . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
Studying the Rasterization Pipeline . . . . . . . . . . . . . . . . . . . . . . . . .476
Light a Ray Tracing Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .483
Light Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489
Lighting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .492
Part V: Optimizations 501
Chapter 19 The Quick Mind: Computational Optimizations . . . . . . . . 503
Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .504
IEEE Floating Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .517
Condition Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .524
Chapter 20 Using SIMD to Move Linear Algebra Functions onto
the Highway . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
Matrix Storage Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538
Matrix Transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .538
Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .540
Vector Length and Normalization . . . . . . . . . . . . . . . . . . . . . . . . . .542
Matrix-Vector Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .544
Matrix-Matrix Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .547
Matrix Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .551
Cross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .555
Matrix Division (Inverse) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .556