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Mathematics for Second Edition
Hardcover, 570 pages, 173 figures This book illustrates the mathematics that a game programmer would need to develop a professional-quality 3D engine. Although the book is geared toward applications in game development, many of the topics appeal to general interests in 3D graphics. It starts at a fairly basic level in areas such as vector geometry and linear algebra, and then progresses to more advanced topics in 3D game programming such as illumination, visibility determination, and collision detection. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure gaps in the theory. The book assumes a working knowledge of trigonometry and calculus, but also includes sections that review the important tools used from these disciplines, such as trigonometric identities, differential equations, and Taylor series. “Great coverage. Best book of its kind.” |
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Second Edition Contents
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Preface What's New in the Second Edition Contents Overview Notational Conventions Chapter 0: The Rendering Pipeline 0.1 Graphics Processors 0.2 Vertex Transformation 0.3 Rasterization and Fragment Operations Chapter 1: Vectors 1.1 Vector Properties 1.2 Dot Products 1.3 Cross Products 1.4 Vector Spaces Chapter 2: Matrices 2.1 Matrix Properties 2.2 Linear Systems 2.3 Matrix Inverses 2.4 Determinants 2.5 Eigenvalues and Eigenvectors 2.6 Diagonalization Chapter 3: Transforms 3.1 Linear Transformations 3.1.1 Orthogonal Matrices 3.1.2 Handedness 3.2 Scaling Transforms 3.3 Rotation Transforms 3.3.1 Rotation About an Arbitrary Axis 3.4 Homogeneous Coordinates 3.4.1 Four-dimensional Transforms 3.4.2 Points and Directions 3.4.3 Geometrical Interpretation of the w-coordinate 3.5 Transforming Normal Vectors 3.6 Quaternions 3.6.1 Quaternion Mathematics 3.6.2 Rotations with Quaternions 3.6.3 Spherical Linear Interpolation Chapter 4: 3D Engine Geometry 4.1 Lines in 3D Space 4.1.1 Distance Between a Point and a Line 4.1.2 Distance Between Two Lines 4.2 Planes in 3D Space 4.2.1 Intersection of a Line and a Plane 4.2.2 Intersection of Three Planes 4.2.3 Transforming Planes 4.3 The View Frustum 4.3.1 Field of View 4.3.2 Frustum Planes 4.4 Perspective Correct Interpolation 4.4.1 Depth Interpolation 4.4.2 Vertex Attribute Interpolation 4.5 Projections 4.5.1 Perspective Projections 4.5.2 Orthographic Projections 4.5.3 Extracting Frustum Planes Chapter 5: Ray Tracing 5.1 Root Finding 5.1.1 Quadratic Polynomials 5.1.2 Cubic Polynomials 5.1.3 Quartic Polynomials 5.1.4 Newton’s Method 5.1.5 Refinement of Reciprocals and Square Roots 5.2 Surface Intersections 5.2.1 Intersection of a Ray and a Triangle 5.2.2 Intersection of a Ray and a Box 5.2.3 Intersection of a Ray and a Sphere 5.2.4 Intersection of a Ray and a Cylinder 5.2.5 Intersection of a Ray and a Torus 5.3 Normal Vector Calculation 5.4 Reflection and Refraction Vectors 5.4.1 Reflection Vector Calculation 5.4.2 Refraction Vector Calculation Chapter 6: Illumination 6.1 RGB Color 6.2 Light Sources 6.2.1 Ambient Light 6.2.2 Directional Light Sources 6.2.3 Point Light Sources 6.2.4 Spot Light Sources 6.3 Diffuse Lighting 6.4 Texture Mapping 6.4.1 Standard Texture Maps 6.4.2 Projective Texture Maps 6.4.3 Cube Texture Maps 6.4.4 Filtering and Mipmaps 6.5 Specular Lighting 6.6 Emission 6.7 Shading 6.7.1 Calculating Normal Vectors 6.7.2 Gouraud Shading 6.7.3 Phong Shading 6.8 Bump Mapping 6.8.1 Bump Map Construction 6.8.2 Tangent Space 6.8.3 Calculating Tangent Vectors 6.8.4 Implementation 6.9 A Physical Reflection Model 6.9.1 Bidirectional Reflectance Distribution Functions 6.9.2 Cook-Torrance Illumination 6.9.3 The Fresnel Factor 6.9.4 The Microfacet Distribution Function 6.9.5 The Geometrical Attenuation Factor 6.9.6 Implementation Chapter 7: Visibility Determination 7.1 Bounding Volume Construction 7.1.1 Principal Component Analysis 7.1.2 Bounding Box Construction 7.1.3 Bounding Sphere Construction 7.1.4 Bounding Ellipsoid Construction 7.1.5 Bounding Cylinder Construction 7.2 Bounding Volume Tests 7.2.1 Bounding Sphere Test 7.2.2 Bounding Ellipsoid Test 7.2.3 Bounding Cylinder Test 7.2.4 Bounding Box Test 7.3 Spatial Partitioning 7.3.1 Octrees 7.3.2 Binary Space Partitioning Trees 7.4 Portal Systems 7.4.1 Portal Clipping 7.4.2 Reduced View Frustums |
Chapter 8: Collision Detection 8.1 Plane Collisions 8.1.1 Collision of a Sphere and a Plane 8.1.2 Collision of a Box and a Plane 8.1.3 Spatial Partitioning 8.2 General Sphere Collisions 8.3 Sliding 8.4 Collision of Two Spheres Chapter 9: Polygonal Techniques 9.1 Depth Value Offset 9.1.1 Projection Matrix Modification 9.1.2 Offset Value Selection 9.1.3 Implementation 9.2 Decal Application 9.2.1 Decal Mesh Construction 9.2.2 Polygon Clipping 9.3 Billboarding 9.3.1 Unconstrained Quads 9.3.2 Constrained Quads 9.3.3 Polyline Quadstrips 9.4 Polygon Reduction 9.5 T-Junction Elimination 9.6 Triangulation Chapter 10: Shadows 10.1 Algorithm Overview 10.2 Infinite View Frustums 10.3 Silhouette Determination 10.4 Shadow Volume Construction 10.5 Determining Cap Necessity 10.6 Rendering Shadow Volumes 10.7 Scissor Optimization Chapter 11: Linear Physics 11.1 Position Functions 11.2 Second-Order Differential Equations 11.2.1 Homogeneous Equations 11.2.2 Nonhomogeneous Equations 11.2.3 Initial Conditions 11.3 Projectile Motion 11.4 Resisted Motion 11.5 Friction Chapter 12: Rotational Physics 12.1 Rotating Environments 12.1.1 Angular Velocity 12.1.2 The Centrifugal Force 12.1.3 The Coriolis Force 12.2 Rigid Body Motion 12.2.1 Center of Mass 12.2.2 Angular Momentum and Torque 12.2.3 The Inertia Tensor 12.2.4 Principal Axes of Inertia 12.3 Oscillatory Motion 12.3.1 Spring Motion 12.3.2 Pendulum Motion Chapter 13: Fluid Simulation 13.1 The Wave Equation 13.2 Approximating Derivatives 13.3 Evaluating Surface Displacement 13.4 Implementation Chapter 14: Numerical Methods 14.1 Linear Systems 14.1.1 Triangular Systems 14.1.2 Gaussian Elimination 14.1.3 LU Decomposition 14.1.4 Error Reduction 14.1.5 Tridiagonal Systems 14.2 Eigenvalues and Eigenvectors 14.3 Ordinary Differential Equations 14.3.1 Euler’s Method 14.3.2 Taylor Series Method 14.3.3 Runge-Kutta Method 14.3.4 Higher-Order Differential Equations Chapter 15: Curves and Surfaces 15.1 Cubic Curves 15.2 Hermite Curves 15.3 Bézier Curves 15.3.1 Cubic Bézier Curves 15.3.2 Bézier Curve Truncation 15.3.3 The de Casteljau Algorithm 15.4 Catmull-Rom Splines 15.5 Cubic Splines 15.6 B-Splines 15.6.1 Uniform B-Splines 15.6.2 B-Spline Globalization 15.6.3 Nonuniform B-Splines 15.6.4 NURBS 15.7 Bicubic Surfaces 15.8 Curvature and Torsion Appendix A: Complex Numbers A.1 Definition A.2 Addition and Multiplication A.3 Conjugates and Inverses A.4 The Euler Formula Appendix B: Trigonometry Reference B.1 Function Definitions B.2 Symmetry and Phase Shifts B.3 Pythagorean Identities B.4 Exponential Identities B.5 Inverse Functions B.6 Laws of Sines and Cosines Appendix C: Coordinate Systems C.1 Cartesian Coordinates C.2 Cylindrical Coordinates C.3 Spherical Coordinates C.4 Generalized Coordinates Appendix D: Taylor Series D.1 Derivation D.2 Power Series D.3 The Euler Formula Appendix E: Answers to Exercises Bibliography Index |
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Errata
- Page 31. In Exercise 1.3, the third point should be (7, −8, 6).
- Page 41. In Equation (2.19), row 2 should be multiplied by 4/17, not −4/3.
- Page 95. In Figure 3.9, the quaternions q1 and q2 should be reversed in both parts (a) and (b).
- Page 276. In Equation (9.8), the minimum value of ε should be twice the given value because the viewport transformation ordinarily cuts the depth value in half when it maps the homogeneous clip space range [-1,1] to the viewport range [0,1]. The value of k in equation (9.11) should be also be doubled.
- Page 337. In Equation (10.37), the denominator should be Nya.