Game Physics (Hardcover)

David H. Eberly


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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.


  • CD-ROM with extensive C++ source code that supports physical simulation; has many illustrative applications for Windows, Linux, and OS X; and is compatible with many game engines—including the Wild Magic engine, for which the complete source code is included.
  • Contains sample applications for shader programs (OpenGL and DirectX), including deformation by vertex displacement, skin and bones for smooth object animation, rippling ocean waves with realistic lighting, refraction effects, Fresnel reflectance, and iridescence.
  • Includes exercises for instructional use and a review of essential mathematics.
  • Content

    1 Introduction
    1.1 A Brief History of the World
    1.2 A Summary of the Topics
    1.3 Examples and Exercises

    2 Basic Concepts from Physics
    2.1 Rigid Body Classification
    2.2 Rigid Body Kinematics
    2.2.1 Single Particle
    2.2.2 Particle Systems and Continuous Materials
    2.3 Newton's Laws
    2.4 Forces
    2.4.1 Gravitational Forces
    2.4.2 Spring Forces
    2.4.3 Friction and Other Dissipative Forces
    2.4.4 Torque
    2.4.5 Equilibrium
    2.5 Momenta
    2.5.1 Linear Momentum
    2.5.2 Angular Momentum
    2.5.3 Center of Mass
    2.5.4 Moments and Products of Inertia
    2.5.5 Mass and Inertia Tensor of a Solid Polyhedron
    2.6 Energy
    2.6.1 Work and Kinetic Energy
    2.6.2 Conservative Forces and Potential Energy.

    3 Rigid Body Motion
    3.1 Newtonian Dynamics.
    3.2 Lagrangian Dynamics.
    3.2.1 Equations of Motion for a Particle
    3.2.2 Time-Varying Frames or Constraints.
    3.2.3 Interpretation of the Equations of Motion.
    3.2.4 Equations of Motion for a System of Particles
    3.2.5 Equations of Motion for a Continuum of Mass
    3.2.6 Examples with Conservative Forces.
    3.2.7 Examples with Dissipative Forces
    3.3 Euler's Equations of Motion.

    4 Deformable Bodies
    4.1 Elasticity, Stress, and Strain.
    4.2 Mass-Spring Systems.
    4.2.1 One-Dimensional Array of Masses
    4.2.2 Two-Dimensional Array of Masses.
    4.2.3 Three-Dimensional Array of Masses.
    4.2.4 Arbitrary Configurations
    4.3 Control Point Deformation
    4.3.1 B-Spline Curves
    4.3.2 NURBS Curves.
    4.3.3 B-Spline Surfaces
    4.3.4 NURBS Surfaces
    4.3.5 Surfaces Built from Curves
    4.4 Free-Form Deformation
    4.5 Implicit Surface Deformation.
    4.5.1 Level Set Extraction.
    4.5.2 Isocurve Extraction in 2D Images
    4.5.3 Isosurface Extraction in 3D Images.

    5 Physics Engines
    5.1 Unconstrained Motion.
    5.1.1 An Illustrative Implementation
    5.1.2 A Practical Implementation.
    5.2 Constrained Motion
    5.2.1 Collision Points.
    5.2.2 Collision Response for Colliding Contact
    5.2.3 Collision Response for Resting Contact
    5.2.4 An Illustrative Implementation
    5.2.5 Lagrangian Dynamics.
    5.3 Collision Detection with Convex Polyhedra.
    5.3.1 The Method of Separating Axes
    5.3.2 Stationary Objects
    5.3.3 Objects Moving with Constant Linear Velocity
    5.3.4 Oriented Bounding Boxes
    5.3.5 Boxes Moving with Constant Linear and Angular Velocity.
    5.4 Collision Culling, Spatial and Temporal Coherence.
    5.4.1 Culling with Bounding Spheres
    5.4.2 Culling with Axis-Aligned Bounding Boxes.
    5.5 Variations

    6 Physics and Shader Programs
    6.1 Introduction
    6.2 Vertex and Pixel Shaders
    6.3 Deformation by Vertex Displacement.
    6.4 Skin and Bones Animation
    6.5 Rippling Ocean Waves.
    6.6 Refraction
    6.7 Fresnel Reflectance
    6.8 Iridescence

    7 Linear Complementarity and Mathematical Programming
    7.1 Linear Programming.
    7.1.1 A Two-Dimensional Example.
    7.1.2 Solution by Pairwise Intersections
    7.1.3 Statement of the General Problem.
    7.1.4 The Dual Problem
    7.2 The Linear Complementarity Problem
    7.2.1 The Lemke-Howson Algorithm
    7.2.2 Zero Constant Terms.
    7.2.3 The Complementary Variable Cannot Leave the Diction
    7.3 Mathematical Programming.
    7.3.1 Karush-Kuhn-Tucker Conditions
    7.3.2 Convex Quadratic Programming
    7.3.3 General Duality Theory
    7.4 Applications
    7.4.1 Distance Calculations.
    7.4.2 Contact Forces.

    8 Differential Equations
    8.1 First-Order Equations.
    8.2 Existence, Uniqueness, and Continuous Dependence.
    8.3 Second-Order Equations
    8.4 General-Order Differential Equations.
    8.5 Systems of Linear Differential Equations
    8.6 Equilibria and Stability.
    8.6.1 Stability for Constant-Coefficient Linear Systems.
    8.6.2 Stability for General Autonomous Systems.

    9 Numerical Methods
    9.1 Euler's Method.
    9.2 Higher-Order Taylor Methods.
    9.3 Methods Via an Integral Formulation.
    9.4 Runge-Kutta Methods.
    9.4.1 Second-Order Methods.
    9.4.2 Third-Order Methods.
    9.4.3 Fourth-Order Method.
    9.5 Multistep Methods
    9.6 Predictor-Corrector Methods.
    9.7 Extrapolation Methods.
    9.7.1 Richardson Extrapolation
    9.7.2 Application to Differential Equations.
    9.7.3 Polynomial Interpolation and Extrapolation.
    9.7.4 Rational Polynomial Interpolation and Extrapolation
    9.7.5 Modified Midpoint Method
    9.7.6 Bulirsch-Stoer Method.
    9.8 Verlet Integration
    9.8.1 Forces without a Velocity Component.
    9.8.2 Forces with a Velocity Component.
    9.8.3 Simulating Drag in the System
    9.8.4 Leap Frog Method
    9.8.5 Velocity Verlet Method.
    9.8.6 Gear's Fifth-Order Predictor-Corrector Method
    9.9 Numerical Stability and its Relationship to Physical Stability
    9.9.1 Stability for Single-Step Methods
    9.9.2 Stability for Multistep Methods
    9.9.3 Choosing a Stable Step Size.
    9.10 Stiff Equations.

    10 Quaternions
    10.1 Rotation Matrices
    10.2 The Classical Approach
    10.2.1 Relationship of Quaternions to Rotations
    10.3 A Linear Algebraic Approach.
    10.4 From Rotation Matrices to Quaternions
    10.4.1 Introduction
    10.4.2 2D Rotations.
    10.4.3 Linearity.
    10.4.4 3D Rotations: Geometry
    10.4.5 4D Rotations.
    10.4.6 3D Rotations: Algebra.
    10.4.7 4D Matrix
    10.4.8 Retrospect, Prospect.
    10.5 Interpolation of Quaternions.
    10.5.1 Spherical Linear Interpolation.
    10.5.2 Spherical Quadratic Interpolation
    10.6 Derivatives of Time-Varying Quaternions

    A Linear Algebra
    A.1 A Review of Number Systems.
    A.1.1 The Integers
    A.1.2 The Rational Numbers.
    A.1.3 The Real Numbers
    A.1.4 The Complex Numbers.
    A.1.5 Fields.
    A.2 Systems of Linear Equations.
    A.2.1 A Closer Look at Two Equations in Two Unknowns.
    A.2.2 Gaussian Elimination and Elementary Row Operations
    A.2.3 Nonsquare Systems of Equations
    A.2.4 The Geometry of Linear Systems
    A.2.5 Numerical Issues
    A.2.6 Iterative Methods for Solving Linear Systems
    A.3 Matrices.
    A.3.1 Some Special Matrices.
    A.3.2 Elementary Row Matrices
    A.3.3 Inverse Matrices.
    A.3.4 Properties of Inverses.
    A.3.5 Construction of Inverses
    A.3.6 LU Decomposition
    A.4 Vector Spaces.
    A.4.1 Definition of a Vector Space.
    A.4.2 Linear Combinations, Spans, and Subspaces.
    A.4.3 Linear Independence and Bases
    A.4.4 Inner Products, Length, Orthogonality, and Projection
    A.4.5 Dot Product, Cross Product, and Triple Products.
    A.4.6 Orthogonal Subspaces.
    A.4.7 The Fundamental Theorem of Linear Algebra
    A.4.8 Projection and Least Squares.
    A.4.9 Linear Transformations.
    A.5 Advanced Topics
    A.5.1 Determinants.
    A.5.2 Eigenvalues and Eigenvectors.
    A.5.3 Eigendecomposition for Symmetric Matrices.
    A.5.4 S + N Decomposition.
    A.5.5 Applications

    B Affine Algebra
    B.1 Introduction
    B.2 Coordinate Systems
    B.3 Subspaces
    B.4 Transformations.
    B.5 Barycentric Coordinates
    B.5.1 Triangles.
    B.5.2 Tetrahedra
    B.5.3 Simplices
    B.5.4 Length, Area, Volume, and Hypervolume

    C Calculus
    C.1 Univariate Calculus
    C.1.1 Limits.
    C.1.2 Limits of a Sequence.
    C.1.3 Continuity
    C.1.4 Differentiation.
    C.1.5 l'Hôpital's Rule.
    C.1.6 Integration
    C.2 Multivariate Calculus.
    C.2.1 Limits and Continuity.
    C.2.2 Differentiation.
    C.2.3 Integration
    C.3 Applications
    C.3.1 Optimization.
    C.3.2 Constrained Optimization
    C.3.3 Derivative Approximations by Finite Differences

    D Ordinary Difference Equations
    D.1 Definitions
    D.2 Linear Equations
    D.2.1 First-Order Linear Equations.
    D.2.2 Second-Order Linear Equations
    D.3 Constant Coefficient Equations
    D.4 Systems of Equations.