Trademarks

Figures

Tables

Preface

About the CD-ROM**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 Algebraic Operations

10.2.2 Relationship of Quaternions to Rotations

10.3 A Linear Algebraic Approach

10.4 From Rotation Matrices to Quaternions

*Contributed by Ken Shoemake*

10.4.1 2D Rotations

10.4.2 Linearity

10.4.3 3D Rotations: Geometry

10.4.4 4D Rotations

10.4.5 3D Rotations: Algebra

10.4.6 4D Matrix

10.4.7 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

Bibliography

Index