GAME PHYSICS SECOND EDITION. дяййтаййг 1 *

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1 GAME PHYSICS SECOND EDITION DAVID H. EBERLY дяййтаййг 1 * К AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO MORGAN ELSEVIER Morgan Kaufmann Publishers is an imprint of Elsevier KAUFMANN

2 CONTENTS TRADEMARKS FIGURES TABLES PREFACE TO THE SECOND EDITION PREFACE TO THE FIRST EDITION ABOUT THE CD-ROM xvu xix xxxiii xxxv xxxvii xli INTRODUCTION 1.1 A BRIEF HISTORY OF THE WORLD 1.2 A SUMMARY OF THE TOPICS 1.3 EXAMPLES AND EXERCISES % BASIC CONCEPTS FROM PHYSICS RIGID BODY CLASSIFICATION RIGID BODY KINEMATICS Single Particle Particle Systems and Continuous Materials NEWTON'S LAWS FORCES Gravitational Forces Spring Forces Friction and Other Dissipative Forces Torque Equilibrium MOMENTA Linear Momentum Angular Momentum Center of Mass vu

3 viii Contents Moments and Products of Inertia Mass and Inertia Tensor of a Solid Polyhedron 2.6 ENERGY Work and Kinetic Energy Conservative Forces and Potential Energy Ъ RIGID BODY MOTION NEWTONIAN LAGRANGIAN DYNAMICS DYNAMICS Equations of Motion for a Particle Time-Varying Frames or Constraints Interpretation of the Equations of Motion Equations of Motion for a System of Particles Equations of Motion for a Continuum of Mass Examples with Conservative Forces Examples with Dissipative Forces 3.3 EULER'S EQUATIONS OF MOTION DEFORMABLE BODIES ELASTICITY, STRESS, AND STRAIN 4.2 MASS-SPRING SYSTEMS One-Dimensional Array of Masses Two-Dimensional Array of Masses Three-Dimensional Array of Masses Arbitrary Configurations 4.3 CONTROL POINT DEFORMATION B-Spline Curves NURBS Curves B-Spline Surfaces NURBS Surfaces Surfaces Built from Curves 4.4 FREE-FORM DEFORMATION 4.5 IMPLICIT SURFACE DEFORMATION Level Set Extraction

4 Contents Isocurve Extraction in 2D Images Isosurface Extraction in 3D Images 206 FLUIDS AND GASES VECTOR CALCULUS Gradient, Directional Derivative, and Total Derivative Vector Fields, Divergence, and Laplacian Curl Line Integrals Surface Integrals and Stokes'Theorem Volume Integrals and the Divergence Theorem Green's Theorem, Laplace's Equation, and Poisson's Equation Vector Field Decomposition STRAIN AND STRESS Strain Tensor Stress Tensor The Relationship Between Strain and Stress CONSERVATION LAWS Conservation ofmass Conservation of Momentum A SIMPLIFIED MODEL FOR FLUID FLOW IMPLEMENTING THE SIMPLIFIED 2D MODEL The Density Equation The Diffusion Term The Advection Term The Source-Sink Term The Total Density Update The Velocity Equations Specialized Boundary Handling IMPLEMENTING THE SIMPLIFIED 3D MODEL VARIATIONS OF THE SIMPLIFIED MODEL Vorticity Confinement and Vortex Particles Separate Pressure Term Omit Diffusion Terms Density and Velocity Dissipation Include Temperature Compressible Flow Obstacles in the Fluid Region 293

5 5.7.8 Moving Boundaries and Multiple Fluids Finding Papers on Fluid Simulation PHYSICS ENGINES THE PHYSICS TICK COLLISION CULLING Culling with Bounding Spheres Culling with Axis-Aligned Bounding Boxes AABB Culling in a Single-Threaded Environment AABB Culling Using a Separate Core of a CPU AABB Culling Using a Specialized Processor TEST-INTERSECTION QUERIES Spheres Capsules Ellipsoids Cylinders COLLISION DETECTION WITH CONVEX POLYHEDRA The Method of Separating Axes Stationary Objects Objects Moving with Constant Linear Velocity Oriented Bounding Boxes Boxes Moving with Constant Linear and Angular Velocity GJK Algorithm UNCONSTRAINED MOTION ACCELERATION-BASED CONSTRAINED MOTION Collision Points Collision Response for Colliding Contact Collision Response for Resting Contact An Illustrative Implementation Lagrangian Dynamics VELOCITY-BASED CONSTRAINED MOTION Constraint on a Particle Constraints on a Particle System Constraint on a Rigid Body Constraints on a Rigid Body System Comments and Variations on the Algorithm VARIATIONS 533

6 Contents xi / LINEAR ALGEBRA A REVIEW OF NUMBER SYSTEMS The Integers The Rational Numbers The Real Numbers The Complex Numbers Fields SYSTEMS OF LINEAR EQUATIONS A Closer Look at Two Equations in Two Unknowns Gaussian Elimination and Elementary Row Operations Nonsquare Systems of Equations The Geometry of Linear Systems Numerical Issues - J Iterative Methods for Solving Linear Systems MATRICES Some Special Matrices Elementary Row Matrices Inverse Matrices Properties of Inverses Construction of Inverses LU Decomposition VECTOR SPACES Definition of a Vector Space Linear Combinations, Spans, and Subspaces Linear Independence and Bases Inner Products, Length, Orthogonality, and Projection Dot Product, Cross Product, and Triple Products Orthogonal Subspaces The Fundamental Theorem of Linear Algebra Projection and Least Squares Linear Transformations ADVANCED TOPICS Determinants Eigenvalues and Eigenvectors Eigendecomposition for Symmetric Matrices S + N Decomposition Applications 652

7 xii Contents AFFINE ALGEBRA INTRODUCTION 8.2 COORDINATE SYSTEMS 8.3 SUBSPACES 8.4 TRANSFORMATIONS 8.5 BARYCENTRIC COORDINATES Triangles Tetrahedra Simplices Length, Area, Volume, and Hypervolume CALCULUS 9.1 UNIVARIATE CALCULUS Limits Limits of a Sequence Continuity Differentiation L'Hopital's Rule Integration 9.2 MULTIVARIATE CALCULUS Limits and Continuity Differentiation Integration 9.3 APPLICATIONS Optimization Constrained Optimization Derivative Approximations by Finite Differences QUATERNIONS 10.1 ROTATION MATRICES 10.2 THE CLASSICAL APPROACH Algebraic Operations Relationship of Quaternions to Rotations

8 Contents XÜi 10.3 A LINEAR ALGEBRAIC APPROACH 10.4 INTERPOLATION OF QUATERNIONS Spherical Linear Interpolation Spherical Quadratic Interpolation 10.5 DERIVATIVES OF TIME-VARYING QUATERNIONS DIFFERENTIAL EQUATIONS FIRST-ORDER EQUATIONS EXISTENCE, UNIQUENESS, AND CONTINUOUS DEPENDENCE SECOND-ORDER EQUATIONS GENERAL-ORDER DIFFERENTIAL EQUATIONS SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS EQUILIBRIA AND STABILITY Stability for Constant-Coefficient Linear Systems Stability for General Autonomous Systems 749 ORDINARY DIFFERENCE EQUATIONS 12.1 DEFINITIONS 12.2 LINEAR EQUATIONS First-Order Linear Equations Second-Order Linear Equations 12.3 CONSTANT COEFFICIENT EQUATIONS 12.4 SYSTEMS OF EQUATIONS NUMERICAL METHODS EULER'S METHOD 13.2 HIGHER-ORDER TAYLOR METHODS 13.3 METHODS VIA AN INTEGRAL FORMULATION

9 xiv Contents 13.4 RuNGE-КиттА METHODS Second-Order Methods Third-Order Methods Fourth-Order Method MULTISTEP METHODS PREDICTOR-CORRECTOR METHODS EXTRAPOLATION METHODS Richardson Extrapolation Application to Differential Equations Polynomial Interpolation and Extrapolation Rational Polynomial Interpolation and Extrapolation Modified Midpoint Method Bulirsch-Stoer Method VERLET INTEGRATION Forces Without a Velocity Component Forces with a Velocity Component Simulating Drag in the System Leapfrog Method Velocity Verlet Method Gear's Fifth-Order Predictor-Corrector Method NUMERICAL STABILITY AND ITS RELATIONSHIP TO PHYSICAL STABILITY Stability for Single-Step Methods Stability for Multistep Methods Choosing a Stable Step Size STIFF EQUATIONS LINEAR COMPLEMENTARITY AND MATHEMATICAL PROGRAMMING LINEAR PROGRAMMING A Two-Dimensional Example Solution by Pairwise Intersections Statement of the General Problem The Dual Problem THE LINEAR COMPLEMENTARITY PROBLEM The Lemke Algorithm Zero Constant Terms The Complementary Variable Cannot Leave the Dictionary 837

10 Contents XV 14.3 MATHEMATICAL PROGRAMMING Karush-Kuhn-Tucker Conditions Convex Quadratic Programming General Duality Theory APPLICATIONS Distance Calculations Contact Forces 855 BIBLIOGRAPHY 857 INDEX 865

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