内容

目次

PrefaceIntroductionThe (Continued) Rise of 3D GamesHow to Read this BookPart I Core MathematicsPart II RenderingPart III AnimationPart IV SimulationAppendicesInteractive Demo ApplicationsSupport LibrariesMath LibrariesEngine and Rendering LibrariesReferences and Further ReadingPart I Core MathematicsChapter 1 Vectors and Points1.1 Introduction1.2 Vectors1.2.1 Vectors as Geometry1.2.2 Real Vector Spaces1.2.3 Linear Combinations and Basis Vectors1.2.4 Basic Vector Class Implementation1.2.5 Vector Length1.2.6 Dot Product1.2.7 Gram-Schmidt Orthogonalization1.2.8 Cross Product1.2.9 Triple Products1.3 Points1.3.1 Points as Geometry1.3.2 Affine Spaces1.3.3 Affine Combinations1.3.4 Point Implementation1.3.5 Polar and Spherical Coordinates1.4 Lines1.4.1 Definition1.4.2 Parameterized Lines1.4.3 Generalized Line Equation1.4.4 Collinear Points1.5 Planes1.5.1 Parameterized Planes1.5.2 Generalized Plane Equation1.5.3 Coplanar Points1.6 Polygons and Triangles1.7 Chapter SummaryChapter 2 Linear Transformations and Matrices2.1 Introduction2.2 Linear Transformations2.2.1 Definitions2.2.2 Null Space and Range2.2.3 Linear Transformations and Basis Vectors2.3 Matrices2.3.1 Introduction to Matrices2.3.2 Simple Operations2.3.3 Vector Representation2.3.4 Block Matrices2.3.5 Matrix Product2.3.6 Transforming Vectors2.3.7 Combining Linear Transformations2.3.8 Identity Matrix2.3.9 Performing Vector Operations with Matrices2.3.10 Implementation2.4 Systems of Linear Equations2.4.1 Definition2.4.2 Solving Linear Systems2.4.3 Gaussian Elimination2.5 Matrix Inverse2.5.1 Definition2.5.2 Simple Inverses2.6 The Determinant2.6.1 Definition2.6.2 Computing the Determinant2.6.3 Determinants and Elementary Row Operations2.6.4 Adjoint Matrix and Inverse2.7 Chapter SummaryChapter 3 Affine Transformations3.1 Introduction3.2 Affine Transformations3.2.1 Definition3.2.2 Representation3.3 Standard Affine Transformations3.3.1 Translation3.3.2 Rotation3.3.3 Scaling3.3.4 Reflection3.3.5 Shear3.3.6 Applying an Affine Transformation Around an Arbitrary Point3.3.7 Transforming Plane Normals3.4 Using Affine Transformations3.4.1 Manipulation of Game Objects3.4.2 Matrix Decomposition3.4.3 Avoiding Matrix Decomposition3.5 Object Hierarchies and Scene Graphs3.5.1 Object Hierarchies3.5.2 Scene Graphs3.6 Chapter SummaryChapter 4 Real-World Computer Number Representation4.1 Introduction4.2 Representing Integral Types on a Computer4.2.1 Finiteness of Representation4.2.2 Range4.3 Representing Real Numbers4.3.1 Approximations4.3.2 Precision and Error4.4 Fixed Point4.4.1 Introduction4.4.2 Basic Representation4.4.3 Range and Precision4.4.4 Addition and Subtraction4.4.5 Multiplication4.4.6 Division4.4.7 Real-World Fixed Point4.4.8 Intermediate Value Overflow and Underflow4.4.9 Limits of Fixed Point4.4.10 Fixed Point Summary4.5 Floating-Point Numbers4.5.1 Review: Scientific Notation4.5.2 A Restricted Scientific Notation4.6 Binary "Scientific Notation"4.7 IEEE Floating Point Standard4.7.1 Basic Representation4.7.2 Range and Precision4.7.3 Arithmetic Operations4.7.4 Special Values4.7.5 Very Small Values4.7.6 Catastrophic Cancellation4.7.7 Double Precision4.8 Real-World Floating Point4.8.1 Internal FPU Precision4.8.2 Performance4.8.3 IEEE Specification Compliance4.9 Code4.10 Chapter SummaryPart II RenderingChapter 5 Viewing and Projection5.1 Introduction5.2 The View Frame and View Transformation5.2.1 Defining a Virtual Camera5.2.2 Controlling the Camera5.2.3 Constructing the View Transformation5.3 Projective Transformation5.3.1 Definition5.3.2 The View Frustum5.3.3 Normalized Device Coordinates5.3.4 Homogeneous Coordinates5.3.5 Perspective Projection5.3.6 Oblique Perspective5.3.7 Orthographic Parallel Projection5.3.8 Oblique Parallel Projection5.4 Culling and Clipping5.4.1 Why Cull or Clip?5.4.2 Culling5.4.3 General Plane Clipping5.4.4 Homogeneous Clipping5.5 Screen Transformation5.6 Picking5.7 Management of Viewing Transformations5.8 Chapter SummaryChapter 6 Geometry, Shading, and Texturing6.1 Introduction6.2 Color Representation6.2.1 The RGB Color Model6.2.2 Colors as "Vectors"6.2.3 Operations on Colors6.2.4 Color Range Limitation6.2.5 Alpha Values6.2.6 Color Storage Formats6.2.7 Colors in OpenGL6.3 Points and Vertices6.3.1 Per-Vertex Attributes6.4 Surface Representation6.4.1 Vertices and Surface Ambiguity6.4.2 Triangles6.4.3 Triangle Attributes6.4.4 Vertex Indices6.4.5 OpenGL Vertex Indices6.5 Coloring a Surface6.6 Using Constant Colors6.6.1 Per-Object Colors6.6.2 Per-Triangle Colors6.6.3 Per-Vertex Colors6.6.4 Limitations of Basic Shading Methods6.7 Texture Mapping6.7.1 Introduction6.7.2 Shading via Image Lookup6.7.3 Texture Images6.8 Texture Coordinates6.8.1 Mapping Texture Coordinates6.8.2 Generating Texture Coordinates6.8.3 Texture Coordinate Discontinuities6.8.4 Mapping Outside the Unit Square6.9 Reviewing the Steps of Texturing6.10 Limitations of Texturing6.11 Procedural Colors and Shaders6.12 Chapter SummaryChapter 7 Lighting7.1 Introduction7.2 Basics of Light Approximation7.2.1 Measuring Light7.2.2 Light as a Ray7.3 Lighting Approximation (OpenGL)7.4 Types of Light Sources7.4.1 Directional Lights7.4.2 Point Lights7.4.3 Spotlights7.4.4 Other Types of Light Sources7.5 Surface Materials and Light Interaction7.5.1 OpenGL Materials7.6 Categories of Light7.6.1 Emission7.6.2 Ambient7.6.3 Diffuse7.6.4 Specular7.7 Combined Lighting Equation7.8 Lighting and Shading7.8.1 Flat-Shaded Lighting7.8.2 Per-Vertex Lighting7.8.3 Per-Pixel Lighting (Phong Shading)7.9 Merging Textures and Lighting7.9.1 Specular Lighting and Textures7.10 Lighting and Programmable Shaders7.11 Chapter SummaryChapter 8 Rasterization8.1 Introduction8.2 Displays and Framebuffers8.2.1 Framebuffer Memory Organization8.2.2 Interlacing8.2.3 Multiple Buffers8.3 Conceptual Rasterization Pipeline8.4 Determining the Pixels Contained by a Triangle8.5 Determining Which Pixels are Visible8.5.1 Depth Sorting8.5.2 Depth Buffering8.5.3 Depth Buffering in OpenGL8.6 Computing Source Pixel Colors8.6.1 Flat Colors8.6.2 Gouraud Colors8.7 Rasterizing Textures8.7.1 Texture Coordinate Review8.7.2 Interpolating Texture Coordinates8.7.3 Mapping a Coordinate to a Texel8.7.4 Mipmapping8.8 Blending8.8.1 Blending and Z-Buffering8.8.2 Alternative Blending Modes8.8.3 Blending and OpenGL8.9 Antialiasing8.9.1 Antialiasing in Practice8.9.2 Antialiasing in OpenGL8.10 Chapter SummaryPart III AnimationChapter 9 Curves9.1 Introduction9.2 General Definitions9.3 Linear Interpolation9.3.1 Definition9.3.2 Piecewise Linear Interpolation9.4 Lagrange Polynomials9.5 Hermite Curves9.5.1 Definition9.5.2 Automatic Generation of Hermite Curves9.5.3 Natural, Cyclic, and Acyclic End Conditions9.6 Catmull-Rom Splines9.7 Bézier Curves9.7.1 Definition9.7.2 Piecewise Bézier Curves9.8 B-Splines9.9 Rational Curves9.10 Rendering Curves9.10.1 Forward Differencing9.10.2 Midpoint Subdivision9.10.3 Using OpenGL9.11 Controlling Speed Along a Curve9.11.1 Moving at Constant Speed9.11.2 Computing Arc Length9.11.3 Ease-In and Ease-Out9.12 Camera Control9.13 Chapter SummaryChapter 10 Orientation Representation10.1 Introduction10.2 Rotation Matrices10.3 Fixed and Euler Angles10.3.1 Definition10.3.2 Format Conversion10.3.3 Concatenation10.3.4 Vector Rotation10.3.5 Other Issues10.4 Axis-Angle Representation10.4.1 Definition10.4.2 Format Conversion10.4.3 Concatenation10.4.4 Vector Rotation10.4.5 Section Summary10.5 Quaternions10.5.1 Definition10.5.2 Rotation Quaternions10.5.3 Format Conversion10.5.4 Addition and Scalar Multiplication10.5.5 Negation10.5.6 Magnitude and Normalization10.5.7 Dot Product10.5.8 Concatenation10.5.9 Identity and Inverse10.5.10 Vector Rotation10.5.11 Quaternions and Transformations10.6 Interpolation10.6.1 Linear Interpolation10.6.2 Spherical Linear Interpolation10.6.3 Performance Improvements10.7 Chapter SummaryPart IV SimulationChapter 11 Intersection Testing11.1 Introduction11.2 Closest Point and Distance Tests11.2.1 Closest Point on Line to Point11.2.2 Line-Point Distance11.2.3 Closest Point on Line Segment to Point11.2.4 Line Segment-Point Distance11.2.5 Closest Points between Two Lines11.2.6 Line-Line Distance11.2.7 Closest Points between Two Line Segments11.2.8 Line Segment-Line Segment Distance11.2.9 General Linear Components11.3 Object Intersection11.3.1 Spheres11.3.2 Axis-Aligned Bounding Boxes11.3.3 Swept Spheres11.3.4 Object-Oriented Boxes11.3.5 Triangles11.4 A Simple Collision System11.4.1 Choosing a Base Primitive11.4.2 Bounding Hierarchies11.4.3 Dynamic Objects11.4.4 Performance Improvements11.4.5 Related Systems11.4.6 Section Summary11.5 Chapter SummaryChapter 12 Rigid Body Dynamics12.1 Introduction12.2 Linear Dynamics12.2.1 Moving with Constant Acceleration12.2.2 Forces12.2.3 Linear Momentum12.2.4 Moving with Variable Acceleration12.3 Initial Value Problems12.3.1 Definition12.3.2 Euler's Method12.3.3 Midpoint Method12.3.4 Higher-Order Methods12.3.5 Verlet Integration12.3.6 Implicit Methods12.4 Rotational Dynamics12.4.1 Definitions12.4.2 Orientation and Angular Velocity12.4.3 Torque12.4.4 Angular Momentum and Inertial Tensor12.4.5 Integrating Rotational Quantities12.5 Collision Response12.5.1 Locating the Point of Collision12.5.2 Linear Collision Response12.5.3 Rotational Collision Response12.5.4 Other Response Techniques12.6 Efficiency12.7 Chapter SummaryAppendix A Trigonometry ReviewA.1 Basic DefinitionsA.1.1 Ratios on the Right TriangleA.1.2 Extending to General AnglesA.2 Properties of TrianglesA.3 Trigonometric IdentitiesA.3.1 Pythagorean IdentitiesA.3.2 Complementary AngleA.3.3 Even-OddA.3.4 Compound AngleA.3.5 Double AngleA.3.6 Half AngleA.4 InversesAppendix B Calculus ReviewB.1 Limits and ContinuityB.1.1 LimitsB.1.2 ContinuityB.2 DerivativesB.2.1 DefinitionB.2.2 Basic DerivativesB.2.3 Derivatives of Transcendental FunctionsB.2.4 Taylor's SeriesB.3 IntegralsB.3.1 DefinitionB.3.2 Evaluating IntegralsB.3.3 Trapezoidal RuleB.3.4 Gaussian QuadratureB.4 Space CurvesBibliographyIndexTrademarksAbout 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