GAME PHYSICS ENGINE DEVELOPMENT

Transcription

1 GAME PHYSICS ENGINE DEVELOPMENT IAN MILLINGTON i > AMSTERDAM BOSTON HEIDELBERG fpf l LONDON. NEW YORK. OXFORD ^. PARIS SAN DIEGO SAN FRANCISCO втс^н Г^ 4.«Mt-fSSKHbe. SINGAPORE. SYDNEY. TOKYO ELSEVIER Morgan Kaufmann is an imprint of Elsevier MORGAN KAUFMANN PUBLISHERS

2 CONTENTS LIST OF FIGURES PREFACE xvi xix A INTRODUCTION I 1.1 WHAT IS GAME PHYSICS? WHAT IS A PHYSICS ENGINE? Advantages of a Physics Engine Weaknesses of a Physics Engine APPROACHES TO PHYSICS ENGINES Types of Object Contact Resolution Impulses and Forces What We're Building THE MATHEMATICS OF PHYSICS ENGINES The Math You Need to Know The Math We'll Review The Math We'll Introduce THE SOURCE CODE IN THIS BOOK How THIS BOOK IS STRUCTURED 11 PART I PARTICLE PHYSICS 13 Zä THE MATHEMATICS OF PARTICLES VECTORS The Handedness of Space 19 Vll

3 viii Contents Vectors and Directions Scalar and Vector Multiplication Vector Addition and Subtraction Multiplying Vectors The Component Product The Scalar Product The Vector Product The Orthonormal Basis CALCULUS Differential Calculus Integral Calculus SUMMARY 42 T H E LAWS OF MOTION A PARTICLE THE FIRST TWO LAWS The First Law The Second Law The Force Equations Adding Mass to Particles Momentum and Velocity The Force of Gravity THE INTEGRATOR The Update Equations The Complete Integrator SUMMARY 54 THE PARTICLE PHYSICS ENGINE BALLISTICS Setting Projectile Properties Implementation FIREWORKS The Fireworks Data The Fireworks Rules The Implementation SUMMARY 66

4 Physics Engine Design ix PART II MASS-AGGREGATE PHYSICS 67 Э ADDING GENERAL FORCES D'ALEMBERT'S PRINCIPLE FORCE GENERATORS Interfaces and Polymorphism Implementation A Gravity Force Generator A Drag Force Generator BUILT-IN GRAVITY AND DAMPING SUMMARY 79 О SPRINGS AND SPRINGLIKE THINGS HOOK'S LAW The Limit of Elasticity Springlike Things SPRINGLIKE FORCE GENERATORS A Basic Spring Generator An Anchored Spring Generator An Elastic Bungee Generator A Buoyancy Force Generator STIFF SPRINGS The Problem of Stiff Springs Faking Stiff Springs SUMMARY 101 HARD CONSTRAINTS ЮЗ 7.1 SIMPLE COLLISION RESOLUTION The Closing Velocity The Coefficient of Restitution The Collision Direction and the Contact Normal Impulses 107

5 x Contents 7.2 COLLISION PROCESSING Collision Detection Resolving Interpenetration Resting Contacts THE CONTACT RESOLVER ALGORITHM Resolution Order Time-Division Engines COLLISIONLIKE THINGS Cables Rods SUMMARY THE MASS-AGGREGATE PHYSICS ENGINE Ш 8.1 OVERVIEW OF THE ENGINE USING THE PHYSICS ENGINE Rope-Bridges and Cables Friction Blob Games SUMMARY 142 PART III RIGID-BODY PHYSICS 143 THE MATHEMATICS OF ROTATIONS ROTATING OBJECTS IN TWO DIMENSIONS The Mathematics of Angles Angular Speed The Origin and the Center of Mass ORIENTATION IN THREE DIMENSIONS Euler Angles Axis-Angle Rotation Matrices Quaternions ANGULAR VELOCITY AND ACCELERATION The Velocity of a Point Angular Acceleration 160

6 Physics Engine Design xi 9.4 IMPLEMENTING THE MATHEMATICS The Matrix Classes Matrix Multiplication The Matrix Inverse and Transpose Converting a Quaternion to a Matrix Transforming Vectors Changing the Basis of a Matrix The Quaternion Class Normalizing Quaternions Combining Quaternions Rotating Updating by the Angular Velocity SUMMARY LAWS OF MOTION FOR RIGID BODIES THE RIGID BODY NEWTON 2 FOR ROTATION Torque The Moment of Inertia The Inertia Tensor in World Coordinates D'ALEMBERT FOR ROTATION Force Generators THE RIGID-BODY INTEGRATION SUMMARY THE RIGID-BODY PHYSICS ENGINE OVERVIEW OF THE ENGINE USING THE PHYSICS ENGINE A Flight Simulator A Sailing Simulator SUMMARY 227

7 xii Contents PART IV COLLISION DETECTION COLLISION DETECTION COLLISION DETECTION PIPELINE COARSE COLLISION DETECTION BOUNDING VOLUMES Hierarchies Building the Hierarchy Sub-Object Hierarchies SPATIAL DATA STRUCTURES Binary Space Partitioning Oct-Trees and Quad-Trees Grids Multi-Resolution Maps SUMMARY GENERATING CONTACTS COLLISION GEOMETRY Primitive Assemblies Generating Collision Geometry CONTACT GENERATION Contact Data Point-Face Contacts Edge-Edge Contacts Edge-Face Contacts Face-Face Contacts Early-Outs PRIMITIVE COLLISION ALGORITHMS Colliding Two Spheres Colliding a Sphere and a Plane Colliding a Box and a Plane Colliding a Sphere and a Box Colliding Two Boxes Efficiency and General Polyhedra SUMMARY 297

8 Physics Engine Design xiii PART V CONTACT PHYSICS COLLISION RESOLUTION IMPULSES AND IMPULSIVE TORQUES Impulsive Torque Rotating Collisions Handling Rotating Collisions COLLISION IMPULSES Change to Contact Coordinates Velocity Change by Impulse Impulse Change by Velocity Calculating the Desired Velocity Change Calculating the Impulse Applying the Impulse RESOLVING INTERPENETRATION Choosing a Resolution Method Implementing Nonlinear Projection Avoiding Excessive Rotation THE COLLISION RESOLUTION PROCESS The Collision Resolution Pipeline Preparing Contact Data Resolving Penetration Resolving Velocity Alternative Update Algorithms SUMMARY RESTING CONTACTS AND FRICTION RESTING FORCES Force Calculations MICRO-COLLISIONS Removing Accelerated Velocity Lowering the Restitution The New Velocity Calculation TYPES OF FRICTION Static and Dynamic Friction Isotropic and Anisotropic Friction IMPLEMENTING FRICTION Friction as Impulses 363

9 xiv Contents Modifying the Velocity Resolution Algorithm Putting It All Together FRICTION AND SEQUENTIAL CONTACT RESOLUTION SUMMARY STABILITY AND OPTIMIZATION STABILITY Quaternion Drift Interpenetration on Slopes Integration Stability The Benefit of Pessimistic Collision Detection Changing Mathematical Accuracy OPTIMIZATIONS Sleep Margins of Error for Penetration and Velocity Contact Grouping Code Optimizations SUMMARY PUTTING IT ALL TOGETHER OVERVIEW OF THE ENGINE USING THE PHYSICS ENGINE Ragdolls Fracture Physics Explosive Physics LIMITATIONS OF THE ENGINE Stacks Reaction Force Friction Joint Assemblies Stiff Springs SUMMARY 419 PART VI WHAT COMES NEXT? OTHER TYPES OF PHYSICS SIMULTANEOUS CONTACT RESOLUTION 423

10 Physics Engine Design xv Thejacobian The Linear Complementary Problem REDUCED COORDINATE APPROACHES SUMMARY 429 APPENDICES COMMON INERTIA TENSORS 431 A.l DISCRETE MASSES 431 A.2 CONTINUOUS MASSES 432 В D A.3 COMMON SHAPES 432 A.3.1 Cuboid 432 A.3.2 Sphere 432 A.3.3 Cylinder 433 A.3.4 Cone 433 USEFUL FRICTION COEFFICIENTS FOR GAMES 434 OTHER PROGRAMMING LANGUAGES 435 C.l С 435 C.2 JAVA 436 C.3 COMMON LANGUAGE RUNTIME (.NET) 436 C.4 LUA 436 MATHEMATICS SUMMARY 438 D.l VECTORS 438 D.2 QUATERNIONS 439 D.3 MATRICES 440 D.4 INTEGRATION 441 D.5 PHYSICS 442 D.6 OTHER FORMULAE 443 BIBLIOGRAPHY 445 INDEX 447