book

3D Math Primer for Graphics and Game Development

by Fletcher Dunn, Ian Parberry

June 2002

Beginner

429 pages

11h 27m

English

Jones & Bartlett Learning

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Cover
Title Page
Copyright
Contents
Acknowledgements
Chapter 1 Introduction
1.1 What is 3D Math?1.2 Why You Should Read This Book1.3 What You Should Know Before Reading This Book1.4 Overview
Chapter 2 The Cartesian Coordinate System
2.1 1D Mathematics2.2 2D Cartesian Mathematics2.2.1 An Example: The Hypothetical City of Cartesia2.2.2 Arbitrary 2D Coordinate Spaces2.2.3 Specifying Locations in 2D Using Cartesian Coordinates2.3 From 2D to 3D2.3.1 Extra Dimension, Extra Axis2.3.2 Specifying Locations in 3D2.3.3 Left-handed vs. Right-handed Coordinate Spaces2.3.4 Some Important Conventions Used in This Book2.4 Exercises
Chapter 3 Multiple Coordinate Spaces
3.1 Why Multiple Coordinate Spaces?3.2 Some Useful Coordinate Spaces3.2.1 World Space3.2.2 Object Space3.2.3 Camera Space3.2.4 Inertial Space3.3 Nested Coordinate Spaces3.4 Specifying Coordinate Spaces3.5 Coordinate Space Transformations3.6 Exercises
Chapter 4 Vectors
4.1 Vector — A Mathematical Definition4.1.1 Vectors vs. Scalars4.1.2 Vector Dimension4.1.3 Notation4.2 Vector — A Geometric Definition4.2.1 What Does a Vector Look Like?4.2.2 Position vs. Displacement4.2.3 Specifying Vectors4.2.4 Vectors as a Sequence of Displacements4.3 Vectors vs. Points4.3.1 Relative Position4.3.2 The Relationship Between Points and Vectors4.4 Exercises
Chapter 5 Operations on Vectors
5.1 Linear Algebra vs. What We Need5.2 Typeface Conventions5.3 The Zero Vector5.4 Negating a Vector5.4.1 Official Linear Algebra Rules5.4.2 Geometric Interpretation5.5 Vector Magnitude (Length)5.5.1 Official Linear Algebra Rules5.5.2 Geometric Interpretation5.6 Vector Multiplication by a Scalar5.6.1 Official Linear Algebra Rules5.6.2 Geometric Interpretation5.7 Normalized Vectors5.7.1 Official Linear Algebra Rules5.7.2 Geometric Interpretation5.8 Vector Addition and Subtraction5.8.1 Official Linear Algebra Rules5.8.2 Geometric Interpretation5.8.3 Vector from One Point to Another5.9 The Distance Formula5.10 Vector Dot Product5.10.1 Official Linear Algebra Rules5.10.2 Geometric Interpretation5.10.3 Projecting One Vector onto Another5.11 Vector Cross Product5.11.1 Official Linear Algebra Rules5.11.2 Geometric Interpretation5.12 Linear Algebra Identities5.13 Exercises

Chapter 6 A Simple 3D Vector Class
6.1 Class Interface6.2 Class Vector3 Definition6.3 Design Decisions6.3.1 Floats vs. Doubles6.3.2 Operator Overloading6.3.3 Provide Only the Most Important Operations6.3.4 Don’t Overload Too Many Operators6.3.5 Use Const Member Functions6.3.6 Use Const & Arguments6.3.7 Member vs. Nonmember Functions6.3.8 No Default Initialization6.3.9 Don’t Use Virtual Functions6.3.10 Don’t Use Information Hiding6.3.11 Global Zero Vector Constant6.3.12 No “point3” Class6.3.13 A Word on Optimization
Chapter 7 Introduction to Matrices
7.1 Matrix — A Mathematical Definition7.1.1 Matrix Dimensions and Notation7.1.2 Square Matrices7.1.3 Vectors as Matrices7.1.4 Transposition7.1.5 Multiplying a Matrix with a Scalar7.1.6 Multiplying Two Matrices7.1.7 Multiplying a Vector and a Matrix7.1.8 Row vs. Column Vectors7.2 Matrix — A Geometric Interpretation7.2.1 How Does a Matrix Transform Vectors?7.2.2 What Does a Matrix Look Like?7.2.3 Summary7.3 Exercises
Chapter 8 Matrices and Linear Transformations
8.1 Transforming an Object vs. Transforming the Coordinate Space8.2 Rotation8.2.1 Rotation in 2D8.2.2 3D Rotation about Cardinal Axes8.2.3 3D Rotation about an Arbitrary Axis8.3 Scale8.3.1 Scaling along Cardinal Axes8.3.2 Scale in an Arbitrary Direction8.4 Orthographic Projection8.4.1 Projecting onto a Cardinal Axis or Plane8.4.2 Projecting onto an Arbitrary Line or Plane8.5 Reflection8.6 Shearing8.7 Combining Transformations8.8 Classes of Transformations8.8.1 Linear Transformations8.8.2 Affine Transformations8.8.3 Invertible Transformations8.8.4 Angle-preserving Transformations8.8.5 Orthogonal Transformations8.8.6 Rigid Body Transformations8.8.7 Summary of Types of Transformations8.9 Exercises
Chapter 9 More on Matrices
9.1 Determinant of a Matrix9.1.1 Official Linear Algebra Rules9.1.2 Geometric Interpretation9.2 Inverse of a Matrix9.2.1 Official Linear Algebra Rules9.2.2 Geometric Interpretation9.3 Orthogonal Matrices9.3.1 Official Linear Algebra Rules9.3.2 Geometric Interpretation9.3.3 Orthogonalizing a Matrix9.4 4×4 Homogenous Matrices9.4.1 4D Homogenous Space9.4.2 4×4 Translation Matrices9.4.3 General Affine Transformations9.4.4 Perspective Projection9.4.5 A Pinhole Camera9.4.6 Perspective Projection Using 4×4 Matrices9.5 Exercises
Chapter 10 Orientation and Angular Displacement in 3D
10.1 What is Orientation?10.2 Matrix Form10.2.1 Which Matrix?10.2.2 Advantages of Matrix Form10.2.3 Disadvantages of Matrix Form10.2.4 Summary10.3 Euler Angles10.3.1 What are Euler Angles?10.3.2 Other Euler Angle Conventions10.3.3 Advantages of Euler Angles10.3.4 Disadvantages of Euler Angles10.3.5 Summary10.4 Quaternions10.4.1 Quaternion Notation10.4.2 Quaternions as Complex Numbers10.4.3 Quaternions as an Axis-Angle Pair10.4.4 Quaternion Negation10.4.5 Identity Quaternion(s)10.4.6 Quaternion Magnitude10.4.7 Quaternion Conjugate and Inverse10.4.8 Quaternion Multiplication (Cross Product)10.4.9 Quaternion “Difference”10.4.10 Quaternion Dot Product10.4.11 Quaternion Log, Exp, and Multiplication by a Scalar10.4.12 Quaternion Exponentiation10.4.13 Quaternion Interpolation — aka “Slerp”10.4.14 Quaternion Splines — aka “Squad”10.4.15 Advantages/Disadvantages of Quaternions10.5 Comparison of Methods10.6 Converting between Representations10.6.1 Converting Euler Angles to a Matrix10.6.2 Converting a Matrix to Euler Angles10.6.3 Converting a Quaternion to a Matrix10.6.4 Converting a Matrix to a Quaternion10.6.5 Converting Euler Angles to a Quaternion10.6.6 Converting a Quaternion to Euler Angles10.7 Exercises
Chapter 11 Transformations in C++
11.1 Overview11.2 Class EulerAngles11.3 Class Quaternion11.4 Class RotationMatrix11.5 Class Matrix4×3
Chapter 12 Geometric Primitives
12.1 Representation Techniques12.1.1 Implicit Form12.1.2 Parametric Form12.1.3 “Straightforward” Forms12.1.4 Degrees of Freedom12.2 Lines and Rays12.2.1 Two Points Representation12.2.2 Parametric Representation of Rays12.2.3 Special 2D Representations of Lines12.2.4 Converting between Representations12.3 Spheres and Circles12.4 Bounding Boxes12.4.1 Representing AABBs12.4.2 Computing AABBs12.4.3 AABBs vs. Bounding Spheres12.4.4 Transforming AABBs12.5 Planes12.5.1 Implicit Definition — The Plane Equation12.5.2 Definition Using Three Points12.5.3 “Best-fit” Plane for More Than Three Points12.5.4 Distance from Point to Plane12.6 Triangles12.6.1 Basic Properties of a Triangle12.6.2 Area of a Triangle12.6.3 Barycentric Space12.6.4 Special Points12.7 Polygons12.7.1 Simple vs. Complex Polygons12.7.2 Self-intersecting Polygons12.7.3 Convex vs. Concave Polygons12.7.4 Triangulation and Fanning12.8 Exercises
Chapter 13 Geometric Tests
13.1 Closest Point on 2D Implicit Line13.2 Closest Point on Parametric Ray13.3 Closest Point on Plane13.4 Closest Point on Circle/Sphere13.5 Closest Point in AABB13.6 Intersection Tests13.7 Intersection of Two Implicit Lines in 2D13.8 Intersection of Two Rays in 3D13.9 Intersection of Ray and Plane13.10 Intersection of AABB and Plane13.11 Intersection of Three Planes13.12 Intersection of Ray and Circle/Sphere13.13 Intersection of Two Circles/Spheres13.14 Intersection of Sphere and AABB13.15 Intersection of Sphere and Plane13.16 Intersection of Ray and Triangle13.17 Intersection of Ray and AABB13.18 Intersection of Two AABBs13.19 Other Tests13.20 Class AABB313.21 Exercises
Chapter 14 Triangle Meshes
14.1 Representing Meshes14.1.1 Indexed Triangle Mesh14.1.2 Advanced Techniques14.1.3 Specialized Representations for Rendering14.1.4 Vertex Caching14.1.5 Triangle Strips14.1.6 Triangle Fans14.2 Additional Mesh Information14.2.1 Texture Mapping Coordinates14.2.2 Surface Normals14.2.3 Lighting Values14.3 Topology and Consistency14.4 Triangle Mesh Operations14.4.1 Piecewise Operations14.4.2 Welding Vertices14.4.3 Detaching Faces14.4.4 Edge Collapse14.4.5 Mesh Decimation14.5 A C++ Triangle Mesh Class
Chapter 15 3D Math for Graphics
15.1 Graphics Pipeline Overview15.2 Setting the View Parameters15.2.1 Specifying the Output Window15.2.2 Pixel Aspect Ratio15.2.3 The View Frustum15.2.4 Field of View and Zoom15.3 Coordinate Spaces15.3.1 Modeling and World Space15.3.2 Camera Space15.3.3 Clip Space15.3.4 Screen Space15.4 Lighting and Fog15.4.1 Math on Colors15.4.2 Light Sources15.4.3 The Standard Lighting Equation — Overview15.4.4 The Specular Component15.4.5 The Diffuse Component15.4.6 The Ambient Component15.4.7 Light Attenuation15.4.8 The Lighting Equation — Putting It All Together15.4.9 Fog15.4.10 Flat Shading and Gouraud Shading15.5 Buffers15.6 Texture Mapping15.7 Geometry Generation/Delivery15.7.1 LOD Selection and Procedural Modeling15.7.2 Delivery of Geometry to the API15.8 Transformation and Lighting15.8.1 Transformation to Clip Space15.8.2 Vertex Lighting15.9 Backface Culling and Clipping15.9.1 Backface Culling15.9.2 Clipping15.10 Rasterization
Chapter 16 Visibility Determination
16.1 Bounding Volume Tests16.1.1 Testing Against the View Frustum16.1.2 Testing for Occlusion16.2 Space Partitioning Techniques16.3 Grid Systems16.4 Quadtrees and Octrees16.5 BSP Trees16.5.1 “Old School” BSPs16.5.2 Arbitrary Dividing Planes16.6 Occlusion Culling Techniques16.6.1 Potentially Visible Sets16.6.2 Portal Techniques
Chapter 17 Afterword
Appendix A Math Review
Summation NotationAngles, Degrees, and RadiansTrig FunctionsTrig Identities
Appendix B References
Index

Content preview from 3D Math Primer for Graphics and Game Development

Chapter 14

Triangle Meshes

This chapter discusses a number of issues related to storing and manipulating triangle meshes. It is divided into five sections.

Section 14.1 discusses a number of different ways to represent triangle meshes in a computer.

Section 14.2 discusses different types of data that we store with triangle meshes, other than the basic geometric information.

Section 14.3 introduces mesh topology.

Section 14.4 covers a number ...

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3D Math Primer for Graphics and Game Development

by Fletcher Dunn, Ian Parberry

Triangle Meshes

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.