Ebook: Mathematics for game developers

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 -- 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 -- 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 -- 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 -- Chapter 21 Kicking the Turtle: Approximating Common and Slow Functions 563 -- Transcendental Function Evaluation 564 -- Physical Model Approximation 584 -- Appendix A Notation and Conventions 591 -- Trigonometric Definitions 593 -- Appendix B Trigonometry 593 -- Symmetry 594 -- Pythagorean Identities 597 -- Exponential Identities 598 -- Cosine and Sine Laws 600 -- Inverse Functions 601 -- Appendix C Integral and Derivative Tables 605