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Full Description
Game Physics is an introduction to the ideas and techniques needed to create physically realistic 3D graphic environments. As a companion volume to Dave Eberly's industry standard 3D Game Engine Design, Game Physics shares a similar practical approach and format. Dave includes simulations to introduce the key problems involved and then gradually reveals the mathematical and physical concepts needed to solve them. He then describes all the algorithmic foundations and uses code examples and working source code to show how they are implemented, culminating in a large collection of physical simulations. This book tackles the complex, challenging issues that other books avoid, including Lagrangian dynamics, rigid body dynamics, impulse methods, resting contact, linear complementarity problems, deformable bodies, mass-spring systems, friction, numerical solution of differential equations, numerical stability and its relationship to physical stability, and Verlet integration methods. Dave even describes when real physics isn't necessary-and hacked physics will do.
Contents
TrademarksFiguresTablesPrefaceAbout the CD-ROM1 Introduction1.1 A Brief History of the World1.2 A Summary of the Topics1.3 Examples and Exercises2 Basic Concepts from Physics2.1 Rigid Body Classification2.2 Rigid Body Kinematics2.2.1 Single Particle2.2.2 Particle Systems and Continuous Materials2.3 Newton's Laws2.4 Forces2.4.1 Gravitational Forces2.4.2 Spring Forces2.4.3 Friction and Other Dissipative Forces2.4.4 Torque2.4.5 Equilibrium2.5 Momenta2.5.1 Linear Momentum2.5.2 Angular Momentum2.5.3 Center of Mass2.5.4 Moments and Products of Inertia2.5.5 Mass and Inertia Tensor of a Solid Polyhedron2.6 Energy2.6.1 Work and Kinetic Energy2.6.2 Conservative Forces and Potential Energy3 Rigid Body Motion3.1 Newtonian Dynamics3.2 Lagrangian Dynamics3.2.1 Equations of Motion for a Particle3.2.2 Time-Varying Frames or Constraints3.2.3 Interpretation of the Equations of Motion3.2.4 Equations of Motion for a System of Particles3.2.5 Equations of Motion for a Continuum of Mass3.2.6 Examples with Conservative Forces3.2.7 Examples with Dissipative Forces3.3 Euler's Equations of Motion4 Deformable Bodies4.1 Elasticity, Stress, and Strain4.2 Mass-Spring Systems4.2.1 One-Dimensional Array of Masses4.2.2 Two-Dimensional Array of Masses4.2.3 Three-Dimensional Array of Masses4.2.4 Arbitrary Configurations4.3 Control Point Deformation4.3.1 B-Spline Curves4.3.2 NURBS Curves4.3.3 B-Spline Surfaces4.3.4 NURBS Surfaces4.3.5 Surfaces Built from Curves4.4 Free-Form Deformation4.5 Implicit Surface Deformation4.5.1 Level Set Extraction4.5.2 Isocurve Extraction in 2D Images4.5.3 Isosurface Extraction in 3D Images5 Physics Engines5.1 Unconstrained Motion5.1.1 An Illustrative Implementation5.1.2 A Practical Implementation5.2 Constrained Motion5.2.1 Collision Points5.2.2 Collision Response for Colliding Contact5.2.3 Collision Response for Resting Contact5.2.4 An Illustrative Implementation5.2.5 Lagrangian Dynamics5.3 Collision Detection with Convex Polyhedra5.3.1 The Method of Separating Axes5.3.2 Stationary Objects5.3.3 Objects Moving with Constant Linear Velocity5.3.4 Oriented Bounding Boxes5.3.5 Boxes Moving with Constant Linear and Angular Velocity5.4 Collision Culling: Spatial and Temporal Coherence5.4.1 Culling with Bounding Spheres5.4.2 Culling with Axis-Aligned Bounding Boxes5.5 Variations6 Physics and Shader Programs6.1 Introduction6.2 Vertex and Pixel Shaders6.3 Deformation by Vertex Displacement6.4 Skin-and-Bones Animation6.5 Rippling Ocean Waves6.6 Refraction6.7 Fresnel Reflectance6.8 Iridescence7 Linear Complementarity and Mathematical Programming7.1 Linear Programming7.1.1 A Two-Dimensional Example7.1.2 Solution by Pairwise Intersections7.1.3 Statement of the General Problem7.1.4 The Dual Problem7.2 The Linear Complementarity Problem7.2.1 The Lemke-Howson Algorithm7.2.2 Zero Constant Terms7.2.3 The Complementary Variable Cannot Leave the Diction7.3 Mathematical Programming7.3.1 Karush-Kuhn-Tucker Conditions7.3.2 Convex Quadratic Programming7.3.3 General Duality Theory7.4 Applications7.4.1 Distance Calculations7.4.2 Contact Forces8 Differential Equations8.1 First-Order Equations8.2 Existence, Uniqueness, and Continuous Dependence8.3 Second-Order Equations8.4 General-Order Differential Equations8.5 Systems of Linear Differential Equations8.6 Equilibria and Stability8.6.1 Stability for Constant-Coefficient Linear Systems8.6.2 Stability for General Autonomous Systems9 Numerical Methods9.1 Euler's Method9.2 Higher-Order Taylor Methods9.3 Methods Via an Integral Formulation9.4 Runge-Kutta Methods9.4.1 Second-Order Methods9.4.2 Third-Order Methods9.4.3 Fourth-Order Method9.5 Multistep Methods9.6 Predictor-Corrector Methods9.7 Extrapolation Methods9.7.1 Richardson Extrapolation9.7.2 Application to Differential Equations9.7.3 Polynomial Interpolation and Extrapolation9.7.4 Rational Polynomial Interpolation and Extrapolation9.7.5 Modified Midpoint Method9.7.6 Bulirsch-Stoer Method9.8 Verlet Integration9.8.1 Forces without a Velocity Component9.8.2 Forces with a Velocity Component9.8.3 Simulating Drag in the System9.8.4 Leap Frog Method9.8.5 Velocity Verlet Method9.8.6 Gear's Fifth-Order Predictor-Corrector Method9.9 Numerical Stability and its Relationship to Physical Stability9.9.1 Stability for Single-Step Methods9.9.2 Stability for Multistep Methods9.9.3 Choosing a Stable Step Size9.10 Stiff Equations10 Quaternions10.1 Rotation Matrices10.2 The Classical Approach10.2.1 Algebraic Operations10.2.2 Relationship of Quaternions to Rotations10.3 A Linear Algebraic Approach10.4 From Rotation Matrices to Quaternions Contributed by Ken Shoemake10.4.1 2D Rotations10.4.2 Linearity10.4.3 3D Rotations: Geometry10.4.4 4D Rotations10.4.5 3D Rotations: Algebra10.4.6 4D Matrix10.4.7 Retrospect, Prospect10.5 Interpolation of Quaternions10.5.1 Spherical Linear Interpolation10.5.2 Spherical Quadratic Interpolation10.6 Derivatives of Time-Varying QuaternionsA Linear AlgebraA.1 A Review of Number SystemsA.1.1 The IntegersA.1.2 The Rational NumbersA.1.3 The Real NumbersA.1.4 The Complex NumbersA.1.5 FieldsA.2 Systems of Linear EquationsA.2.1 A Closer Look at Two Equations in Two UnknownsA.2.2 Gaussian Elimination and Elementary Row OperationsA.2.3 Nonsquare Systems of EquationsA.2.4 The Geometry of Linear SystemsA.2.5 Numerical IssuesA.2.6 Iterative Methods for Solving Linear SystemsA.3 MatricesA.3.1 Some Special MatricesA.3.2 Elementary Row MatricesA.3.3 Inverse MatricesA.3.4 Properties of InversesA.3.5 Construction of InversesA.3.6 LU DecompositionA.4 Vector SpacesA.4.1 Definition of a Vector SpaceA.4.2 Linear Combinations, Spans, and SubspacesA.4.3 Linear Independence and BasesA.4.4 Inner Products, Length, Orthogonality, and ProjectionA.4.5 Dot Product, Cross Product, and Triple ProductsA.4.6 Orthogonal SubspacesA.4.7 The Fundamental Theorem of Linear AlgebraA.4.8 Projection and Least SquaresA.4.9 Linear TransformationsA.5 Advanced TopicsA.5.1 DeterminantsA.5.2 Eigenvalues and EigenvectorsA.5.3 Eigendecomposition for Symmetric MatricesA.5.4 S + N DecompositionA.5.5 ApplicationsB Affine AlgebraB.1 IntroductionB.2 Coordinate SystemsB.3 SubspacesB.4 TransformationsB.5 Barycentric CoordinatesB.5.1 TrianglesB.5.2 TetrahedraB.5.3 SimplicesB.5.4 Length, Area, Volume, and HypervolumeC CalculusC.1 Univariate CalculusC.1.1 LimitsC.1.2 Limits of a SequenceC.1.3 ContinuityC.1.4 DifferentiationC.1.5 L'Hopital's RuleC.1.6 IntegrationC.2 Multivariate CalculusC.2.1 Limits and ContinuityC.2.2 DifferentiationC.2.3 IntegrationC.3 ApplicationsC.3.1 OptimizationC.3.2 Constrained OptimizationC.3.3 Derivative Approximations by Finite DifferencesD Ordinary Difference EquationsD.1 DefinitionsD.2 Linear EquationsD.2.1 First-Order Linear EquationsD.2.2 Second-Order Linear EquationsD.3 Constant Coefficient EquationsD.4 Systems of EquationsBibliographyIndex
