Mathematics and Physics for Programmers

The casual game market continues to expand, and with the use of Flash growing and the increased popularity of the iPhone and other mobile gaming devices, developers have an exciting new route to market. But many of these budding game programmers find that they're missing the core math and physics skills they need to develop even simple games. Mathematics and Physics for Programmers (Second Edition) teaches beginning game programmers they skills they need to develop causal games. Beginning with the basic aspects of mathematics and physics that are relevant to games, and moving on to more complex topics, the book combines theory and practice and the reader develops a number of games throughout the course of the book. With the concepts and techniques presented in the book, readers should be able to program a number of standard casual game types, including Mario-style platformers, driving simulations, puzzle games, and a simple first-person shooter. They will also gain a basic understanding of 2D and 3D geometry. Features From the Publisher Provides coverage of all the latest casual gaming technology, including mobile devices such as the iPhone. Written for beginning game programmers who need to learn the basic mathematics and physics skills necessary to develop casual games. The book covers both theory and practice through hands-on application of new skills learned. All examples and techniques are non-platform specific. An accompanying CD-ROM includes all the source code needed to create the games in the book. Cover 1 Table of Contents 8 Part I: Essential Topics in Mathematics 22 Chapter 1 Numbers 24 Overview 24 Numbers as Written 25 How Computers Represent Numbers 30 Exercises 41 Summary 42 You Should Now Know 42 Chapter 2 Arithmetic 44 Overview 44 Fractions 45 Proportions, Ratios, and Percentages 56 Exponentials 64 Logarithms 68 Exercises 72 Summary 73 You Should Now Know 73 Chapter 3 Algebra 74 Overview 74 Basic Algebra 74 Working with Equations 80 Factoring and Solving Quadratic Equations 85 Functions and Graphs 97 Exercises 108 Summary 109 You Should Now Know 109 Chapter 4 Geometry and Trigonometry 110 Overview 110 Angles 110 Triangles 115 Calculations with Triangles 126 Rotations and Reflections 131 Exercises 139 Summary 139 You Should Now Know 140 Chapter 5 Vectors 142 Overview 142 Getting from Here to There 142 Vector Motion 151 Vector Calculations 159 Matrices 168 Exercises 176 Summary 176 You Should Now Know 177 Chapter 6 Calculus 178 Overview 178 Differentiation and Integration 178 Differential Equations 191 Approximation Methods 194 Exercises 200 Summary 201 You Should Now Know 201 Part II: Essential Topics in Physics 202 Chapter 7 Acceleration, Mass, and Energy 204 Overview 204 Ballistics 204 Mass and Momentum 211 Energy 213 Exercises 217 Summary 218 You Should Now Know 219 Chapter 8 Detecting Collisions Between Simple Shapes 220 Overview 220 Ground Rules 221 When Circles Collide 222 When Squares Collide 232 When Ellipses Collide 241 When Things Collide 247 Exercises 248 Summary 249 You Should Now Know 249 Chapter 9 Collision Resolution 250 Overview 250 Resolving a Single Collision 251 Multiple Collisions 259 Exercises 263 Summary 264 You Should Now Know 264 Chapter 10 Detecting Collisions Between Complex Shapes 266 Overview 266 Problems with Complex Shapes 267 Some Reasonable Problems 280 Built-In Solutions 294 Exercises 295 Summary 295 You Should Now Know 296 Chapter 11 A Simple Pool Game 298 Overview 298 Primary Elements of a Simulation 299 Taking a Shot 306 Exercises 314 Summary 315 You Should Now Know 315 Part III: Complex Motion 316 Chapter 12 Force and Newton’s Laws 318 Overview 318 Force 319 Gravity 323 Rockets and Satellites 326 Exercise 329 Summary 329 You Should Now Know 329 Chapter 13 Angular Motion 330 Overview 330 The Physics of a Lever 330 Spin 337 Spinning Collisions 340 Spin Applied to the Pool Game 355 Exercises 356 Summary 357 You Should Now Know 357 Chapter 14 Friction 358 Overview 358 How Friction Works 359 Friction and Angular Motion 364 Exercise 368 Summary 368 You Should Now Know 369 Chapter 15 Strings, Pulleys, and Conveyor Belts 370 Overview 370 Pulling Things Around 370 Continuous Momentum 376 Exercise 380 Summary 380 You Should Now Know 380 Chapter 16 Oscillations 382 Overview 382 Springs 383 Simple Harmonic Motion 385 Damped Harmonic Motion 390 Complications of Springs 395 Calculating Spring Motion 397 Waves 403 Exercise 411 Summary 411 You Should Now Know 411 Part IV: 3-D Mathematics 412 Chapter 17 3-D Geometry 414 Overview 414 3-D Vectors 414 Rendering 423 Casting a Ray 430 Exercise 435 Summary 435 You Should Now Know 435 Chapter 18 Transforms 436 Overview 436 Describing Locations in Space 436 Applying Transforms 442 Exercise 451 Summary 451 You Should Now Know 451 Chapter 19 3-D Collision Detection 452 Overview 452 Colliding Worlds 453 Colliding Footballs 456 Colliding Boxes 458 Colliding Cans 463 Varieties of Collisions 469 Resolving Collisions in Three Dimensions 470 Exercise 470 Summary 471 You Should Now Know 471 Chapter 20 Lighting and Textures 472 Overview 472 Light 472 Materials 476 Shading 489 Exercises 491 Summary 492 You Should Now Know 492 Chapter 21 Modeling Techniques 494 Overview 494 Mathematical 3-D Modeling 495 Animated Surfaces 504 Bone Animations 508 Exercises 514 Summary 515 You Should Now Know 515 Part V: Game Algorithms 516 Chapter 22 Speeding Things Up 518 Overview 518 Cheap and Expensive Calculations 518 Pseudo-Physics 525 Culling 528 Exercises 535 Summary 535 You Should Now Know 535 Chapter 23 Tile-Based Games 536 Overview 536 Generating a Game from Bits 537 Advanced Tiling 544 Exercises 550 Summary 550 You Should Now Know 550 Chapter 24 Mazes 552 Overview 552 Classifying Mazes 552 Creating Mazes 558 Navigating Within Mazes 568 Exercises 577 Summary 577 You Should Now Know 578 Chapter 25 Game Theory and Artificial Intelligence 580 Overview 580 Introduction to Game Theory 581 Tactical AI 593 Top-Down AI 597 Bottom-Up AI 601 Exercises 607 Summary 608 You Should Now Know 608 Chapter 26 Search Techniques 610 Overview 610 Problem Solving 611 Case Study 615 Genetic Algorithms 619 Exercises 626 Summary 626 You Should Now Know 627 Appendix A: Glossary of Mathematical Terms 628 Appendix B: Code References 650 Appendix C: The Greek Alphabet 654 Appendix D: Learning Resources 656 Appendix E: Answers to Exercises 660 Index 668 A 668 B 670 C 671 D 673 E 674 F 675 G 677 H 678 I 678 J 679 K 679 L 679 M 679 N 681 O 681 P 682 Q 683 R 683 S 684 T 686 U 687 V 687 W 688 Z 688 The casual game market continues to expand, and with the use of Flash growing and the increased popularity of the iPhone and other mobile gaming devices, developers have an exciting new route to market. But many of these budding game programmers find that they're missing the core math and physics skills they need to develop even simple games. "Mathematics & Physics For Programmers, Second Edition" teaches beginning game programming students the skills they need to develop causal games. Beginning with the basic aspects of mathematics and physics that are relevant to games, and moving on to more complex topics, the book combines theory and practice and the student develops a number of games throughout the course of the book