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3D Math Primer for Graphics and Game Development, 2nd Edition

by Fletcher Dunn, Ian Parberry

November 2011

Beginner

846 pages

21h 33m

English

A K Peters/CRC Press

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Preliminaries
Dedication
Epigraph
Acknowledgments
Introduction
Who Should Read This BookWhy You Should Read This BookWhat You Should Know before Reading This BookOverviewFind a Bug in This Book?
Chapter 1 Cartesian Coordinate Systems
1.1 1D Mathematics1.2 2D Cartesian Space1.2.1 An Example: The Hypothetical City of Cartesia1.2.2 Arbitrary 2D Coordinate Spaces1.2.3 Specifying Locations in 2D Using Cartesian Coordinates1.3 3D Cartesian Space1.3.1 Extra Dimension, Extra Axis1.3.2 Specifying Locations in 3D1.3.3 Left-handed versus Right-handed Coordinate Spaces1.3.4 Some Important Conventions Used in This Book1.4 Odds and Ends1.4.1 Summation and Product Notation1.4.2 Interval Notation1.4.3 Angles, Degrees, and Radians1.4.4 Trig Functions1.4.5 Trig Identities1.5 ExercisesFigure 1.1Figure 1.2Figure 1.3Figure 1.4Figure 1.5Figure 1.6Figure 1.7Figure 1.8Figure 1.9Figure 1.10Figure 1.11Figure 1.12Figure 1.13Figure 1.14Figure 1.15Figure 1.16Figure 1.17Figure 1.18Figure 1.19Figure 1.20Figure 1.21Table 1.1Table 1.2
Chapter 2 Vectors
2.1 Mathematical Definition of Vector, and Other Boring Stuff2.2 Geometric Definition of Vector2.3 Specifying Vectors with Cartesian Coordinates2.3.1 Vector as a Sequence of Displacements2.3.2 The Zero Vector2.4 Vectors versus Points2.4.1 Relative Position2.4.2 The Relationship between Points and Vectors2.4.3 It’s All Relative2.5 Negating a Vector2.5.1 Official Linear Algebra Rules2.5.2 Geometric Interpretation2.6 Vector Multiplication by a Scalar2.6.1 Official Linear Algebra Rules2.6.2 Geometric Interpretation2.7 Vector Addition and Subtraction2.7.1 Official Linear Algebra Rules2.7.2 Geometric Interpretation2.7.3 Displacement Vector from One Point to Another2.8 Vector Magnitude (Length)2.8.1 Official Linear Algebra Rules2.8.2 Geometric Interpretation2.9 Unit Vectors2.9.1 Official Linear Algebra Rules2.9.2 Geometric Interpretation2.10 The Distance Formula2.11 Vector Dot Product2.11.1 Official Linear Algebra Rules2.11.2 Geometric Interpretation2.12 Vector Cross Product2.12.1 Official Linear Algebra Rules2.12.2 Geometric Interpretation2.13 Linear Algebra Identities2.14 ExercisesFigure 2.1Figure 2.2Figure 2.3Figure 2.4Figure 2.5Figure 2.6Figure 2.7Figure 2.8Figure 2.9Figure 2.10Figure 2.11Figure 2.12Figure 2.13Figure 2.14Figure 2.15Figure 2.16Figure 2.17Figure 2.18Figure 2.19Figure 2.20Figure 2.21Figure 2.22Figure 2.23Figure 2.24Figure 2.25Figure 2.26Figure 2.27Figure 2.28Figure 2.29Figure 2.30Table 2.1Table 2.2
Chapter 3 Multiple Coordinate Spaces
3.1 Why Bother with Multiple Coordinate Spaces?3.2 Some Useful Coordinate Spaces3.2.1 World Space3.2.2 Object Space3.2.3 Camera Space3.2.4 Upright Space3.3 Basis Vectors and Coordinate Space Transformations3.3.1 Dual Perspectives3.3.2 Specifying Coordinate Spaces3.3.3 Basis Vectors3.4 Nested Coordinate Spaces3.5 In Defense of Upright Space3.6 ExercisesFigure 3.1Figure 3.2Figure 3.3Figure 3.4Figure 3.5Figure 3.6Figure 3.7Figure 3.8Figure 3.9Figure 3.10Figure 3.11Figure 3.12Figure 3.13Table 3.1
Chapter 4 Introduction to Matrices
4.1 Mathematical Definition of Matrix4.1.1 Matrix Dimensions and Notation4.1.2 Square Matrices4.1.3 Vectors as Matrices4.1.4 Matrix Transposition4.1.5 Multiplying a Matrix with a Scalar4.1.6 Multiplying Two Matrices4.1.7 Multiplying a Vector and a Matrix4.1.8 Row versus Column Vectors4.2 Geometric Interpretation of Matrix4.3 The Bigger Picture of Linear Algebra4.4 ExercisesFigure 4.1Figure 4.2Figure 4.3Figure 4.4Figure 4.5
Chapter 5 Matrices and Linear Transformations
5.1 Rotation5.1.1 Rotation in 2D5.1.2 3D Rotation about Cardinal Axes5.1.3 3D Rotation about an Arbitrary Axis5.2 Scale5.2.1 Scaling along the Cardinal Axes5.2.2 Scaling in an Arbitrary Direction5.3 Orthographic Projection5.3.1 Projecting onto a Cardinal Axis or Plane5.3.2 Projecting onto an Arbitrary Line or Plane5.4 Reflection5.5 Shearing5.6 Combining Transformations5.7 Classes of Transformations5.7.1 Linear Transformations5.7.2 Affine Transformations5.7.3 Invertible Transformations5.7.4 Angle-Preserving Transformations5.7.5 Orthogonal Transformations5.7.6 Rigid Body Transformations5.7.7 Summary of Types of Transformations5.8 ExercisesFigure 5.1Figure 5.2Figure 5.3Figure 5.4Figure 5.5Figure 5.6Figure 5.7Figure 5.8Figure 5.9Figure 5.10Table 5.1

Chapter 6 More on Matrices
6.1 Determinant of a Matrix6.1.1 Determinants of 2 × 2 and 3 × 3 matrices6.1.2 Minors and Cofactors6.1.3 Determinants of Arbitrary n × n Matrices6.1.4 Geometric Interpretation of Determinant6.2 Inverse of a Matrix6.2.1 The Classical Adjoint6.2.2 Matrix Inverse—Official Linear Algebra Rules6.2.3 Matrix Inverse—Geometric Interpretation6.3 Orthogonal Matrices6.3.1 Orthogonal Matrices—Official Linear Algebra Rules6.3.2 Orthogonal Matrices—Geometric Interpretation6.3.3 Orthogonalizing a Matrix6.4 4 × 4 Homogeneous Matrices6.4.1 4D Homogeneous Space6.4.2 4 × 4 Translation Matrices6.4.3 General Affine Transformations6.5 4 × 4 Matrices and Perspective Projection6.5.1 A Pinhole Camera6.5.2 Perspective Projection Matrices6.6 ExercisesFigure 6.1Figure 6.2Figure 6.3Figure 6.4Figure 6.5Figure 6.6Figure 6.7Figure 6.8Figure 6.9
Chapter 7 Polar Coordinate Systems
7.1 2D Polar Space7.1.1 Locating Points by Using 2D Polar Coordinates7.1.2 Aliasing7.1.3 Converting between Cartesian and Polar Coordinates in 2D7.2 Why Would Anybody Use Polar Coordinates?7.3 3D Polar Space7.3.1 Cylindrical Coordinates7.3.2 Spherical Coordinates7.3.3 Some Polar Conventions Useful in 3D Virtual Worlds7.3.4 Aliasing of Spherical Coordinates7.3.5 Converting between Spherical and Cartesian Coordinates7.4 Using Polar Coordinates to Specify Vectors7.5 ExercisesFigure 7.1Figure 7.2Figure 7.3Figure 7.4Figure 7.5Figure 7.6Figure 7.7Figure 7.8Listing 7.1Listing 7.2Listing 7.3Listing 7.4
Chapter 8 Rotation in Three Dimensions
8.1 What Exactly is “Orientation”?8.2 Matrix Form8.2.1 Which Matrix?8.2.2 Direction Cosines Matrix8.2.3 Advantages of Matrix Form8.2.4 Disadvantages of Matrix Form8.2.5 Summary of Matrix Form8.3 Euler Angles8.3.1 What Are Euler Angles?8.3.2 Other Euler Angle Conventions8.3.3 Advantages of Euler Angles8.3.4 Disadvantages of Euler Angles8.3.5 Summary of Euler Angles8.4 Axis-Angle and Exponential Map Representations8.5 Quaternions8.5.1 Quaternion Notation8.5.2 What Do Those Four Numbers Mean?8.5.3 Quaternion Negation8.5.4 Identity Quaternion(s)8.5.5 Quaternion Magnitude8.5.6 Quaternion Conjugate and Inverse8.5.7 Quaternion Multiplication8.5.8 Quaternion “Difference”8.5.9 Quaternion Dot Product8.5.10 Quaternion log, exp, and Multiplication by a Scalar8.5.11 Quaternion Exponentiation8.5.12 Quaternion Interpolation, a.k.a. Slerp8.5.13 Advantages and Disadvantages of Quaternions8.5.14 Quaternions as Complex Numbers8.5.15 Summary of Quaternions8.6 Comparison of Methods8.7 Converting between Representations8.7.1 Converting Euler Angles to a Matrix8.7.2 Converting a Matrix to Euler angles8.7.3 Converting a Quaternion to a Matrix8.7.4 Converting a Matrix to a Quaternion8.7.5 Converting Euler Angles to a Quaternion8.7.6 Converting a Quaternion to Euler Angles8.8 ExercisesFigure 8.1Figure 8.2Figure 8.3Figure 8.4Figure 8.5Figure 8.6Figure 8.7Figure 8.8Figure 8.9Figure 8.10Figure 8.11Figure 8.12Figure 8.13Table 8.1Table 8.2Listing 8.1Listing 8.2Listing 8.3Listing 8.4Listing 8.5Listing 8.6Listing 8.7
Chapter 9 Geometric Primitives
9.1 Representation Techniques9.2 Lines and Rays9.2.1 Rays9.2.2 Special 2D Representations of Lines9.2.3 Converting between Representations9.3 Spheres and Circles9.4 Bounding Boxes9.4.1 Representing AABBs9.4.2 Computing AABBs9.4.3 AABBs versus Bounding Spheres9.4.4 Transforming AABBs9.5 Planes9.5.1 The Plane Equation: An Implicit Definition of a Plane9.5.2 Defining a Plane by Using Three Points9.5.3 “Best Fit” Plane for More than Three Points9.5.4 Distance from Point to Plane9.6 Triangles9.6.1 Notation9.6.2 Area of a Triangle9.6.3 Barycentric Space9.6.4 Calculating Barycentric Coordinates9.6.5 Special Points9.7 Polygons9.7.1 Simple versus Complex Polygons9.7.2 Convex versus Concave Polygons9.7.3 Triangulation and Fanning9.8 ExercisesFigure 9.1Figure 9.2Figure 9.3Figure 9.4Figure 9.5Figure 9.6Figure 9.7Figure 9.8Figure 9.9Figure 9.10Figure 9.11Figure 9.12Figure 9.13Figure 9.14Figure 9.15Figure 9.16Figure 9.17Figure 9.18Figure 9.19Figure 9.20Figure 9.21Figure 9.22Figure 9.23Figure 9.24Figure 9.25Figure 9.26Figure 9.27Figure 9.28Figure 9.29Figure 9.30Listing 9.1Listing 9.2Listing 9.3Listing 9.4Listing 9.5Listing 9.6Listing 9.7
Chapter 10 Mathematical Topics from 3D Graphics
10.1 How Graphics Works10.1.1 The Two Major Approaches to Rendering10.1.2 Describing Surface Properties: The BRDF10.1.3 A Very Brief Introduction to Colorimetry and Radiometry10.1.4 The Rendering Equation10.2 Viewing in 3D10.2.1 Specifying the Output Window10.2.2 Pixel Aspect Ratio10.2.3 The View Frustum10.2.4 Field of View and Zoom10.2.5 Orthographic Projection10.3 Coordinate Spaces10.3.1 Model, World, and Camera Space10.3.2 Clip Space and the Clip Matrix10.3.3 The Clip Matrix: Preparing for Projection10.3.4 The Clip Matrix: Applying Zoom and Preparing for Clipping10.3.5 Screen Space10.3.6 Summary of Coordinate Spaces10.4 Polygon Meshes10.4.1 Indexed Triangle Mesh10.4.2 Surface Normals10.5 Texture Mapping10.6 The Standard Local Lighting Model10.6.1 The Standard Lighting Equation: Overview10.6.2 The Specular Component10.6.3 The Diffuse Component10.6.4 The Ambient and Emmissive Components10.6.5 The Lighting Equation: Putting It All Together10.6.6 Limitations of the Standard Model10.6.7 Flat and Gouraud Shading10.7 Light Sources10.7.1 Standard Abstract Light Types10.7.2 Light Attenuation10.7.3 Doom-style Volumetric Lights10.7.4 Precalculated Lighting10.8 Skeletal Animation10.9 Bump Mapping10.9.1 Tangent Space10.9.2 Calculating Tangent Space Basis Vectors10.10 The Real-Time Graphics Pipeline10.10.1 Buffers10.10.2 Delivering the Geometry10.10.3 Vertex-Level Operations10.10.4 Clipping10.10.5 Backface Culling10.10.6 Rasterization, Shading, and Output10.11 Some HLSL Examples10.11.1 Decal Shading and HLSL Basics10.11.2 Basic Per-Pixel Blinn-Phong Lighting10.11.3 Gouraud Shading10.11.4 Bump Mapping10.11.5 Skinned Mesh10.12 Further Reading10.13 ExercisesFigure 10.1Figure 10.2Figure 10.3Figure 10.4Figure 10.5Figure 10.6Figure 10.7Figure 10.8Figure 10.9Figure 10.10Figure 10.11Figure 10.12Figure 10.13Figure 10.14Figure 10.15Figure 10.16Figure 10.17Figure 10.18Figure 10.19Figure 10.20Figure 10.21Figure 10.22Figure 10.23Figure 10.24Figure 10.25Figure 10.26Figure 10.27Figure 10.28Figure 10.29Figure 10.30Figure 10.31Figure 10.32Figure 10.33Figure 10.34Figure 10.35Figure 10.36Figure 10.37Figure 10.38Figure 10.39Figure 10.40Figure 10.41Figure 10.42Figure 10.43Figure 10.44Table 10.1Table 10.2Table 10.3Table 10.4Table 10.5Table 10.6Listing 10.1Listing 10.2Listing 10.3Listing 10.4Listing 10.5Listing 10.6Listing 10.7Listing 10.8Listing 10.9Listing 10.10Listing 10.11Listing 10.12Listing 10.13Listing 10.14Listing 10.15Listing 10.16Listing 10.17Listing 10.18Listing 10.19Listing 10.20Listing 10.21Listing 10.22Listing 10.23Listing 10.24
Chapter 11 Mechanics 1: Linear Kinematics and Calculus
11.1 Overview and Other Expectation-Reducing Remarks11.1.1 What is Left Out?11.1.2 Some Helpful Lies about Our Universe11.2 Basic Quantities and Units11.3 Average Velocity11.4 Instantaneous Velocity and the Derivative11.4.1 Limit Arguments and the Definition of the Derivative11.4.2 Examples of Derivatives11.4.3 Calculating Derivatives from the Definition11.4.4 Notations for the Derivative11.4.5 A Few Differentiation Rules and Shortcuts11.4.6 Derivatives of Some Special Functions with Taylor Series11.4.7 The Chain Rule11.5 Acceleration11.6 Motion under Constant Acceleration11.7 The Integral11.7.1 Examples of Integrals11.7.2 The Relationship between the Derivative and the Integral11.7.3 Summary of Calculus11.8 Uniform Circular Motion11.8.1 Uniform Circular Motion in the Plane11.8.2 Uniform Circular Motion in Three Dimensions11.9 ExercisesFigure 11.1Figure 11.2Figure 11.3Figure 11.4Figure 11.5Figure 11.6Figure 11.7Figure 11.8Figure 11.9Figure 11.10Figure 11.11Figure 11.12Figure 11.13Figure 11.14Figure 11.15Figure 11.16Figure 11.17Table 11.1Table 11.2Table 11.3
Chapter 12 Mechanics 2: Linear and Rotational Dynamics
12.1 Newton’s Three Laws12.1.1 Newton’s First Two Laws: Force and Mass12.1.2 Inertial Reference Frames12.1.3 Newton’s Third Law12.2 Some Simple Force Laws12.2.1 Gravitational Force12.2.2 Frictional Forces12.2.3 Spring Forces12.3 Momentum12.3.1 Conservation of Momentum12.3.2 The Center of Mass12.4 Impulsive Forces and Collisions12.4.1 Perfectly Inelastic Collisions12.4.2 General Collision Response12.4.3 The Dirac Delta12.5 Rotational Dynamics12.5.1 Rotational Kinematics12.5.2 2D Rotational Dynamics12.5.3 3D Rotational Dynamics12.5.4 Collision Response with Rotations12.6 Real-Time Rigid Body Simulators12.6.1 Physics Engine State Variables12.6.2 HighLevel Overview12.6.3 Euler Integration12.6.4 Integration of Rotation12.7 Suggested Reading12.8 ExercisesFigure 12.1Figure 12.2Figure 12.3Figure 12.4Figure 12.5Figure 12.6Figure 12.7Figure 12.8Figure 12.9Figure 12.10Figure 12.11Figure 12.12Figure 12.13Figure 12.14Figure 12.15Figure 12.16Figure 12.17Figure 12.18Figure 12.19Figure 12.20Figure 12.21Figure 12.22Table 12.1Table 12.2Listing 12.1Listing 12.2Listing 12.3Listing 12.4Listing 12.5Listing 12.6Listing 12.7
Chapter 13 Curves in 3D
13.1 Parametric Polynomial Curves13.1.1 Parametric Curves13.1.2 Polynomial Curves13.1.3 Matrix Notation13.1.4 Two Trivial Types of Curves13.1.5 Endpoints in Monomial Form13.1.6 Velocities and Tangents13.2 Polynomial Interpolation13.2.1 Aitken’s Algorithm13.2.2 Lagrange Basis Polynomials13.2.3 Polynomial Interpolation Summary13.3 Hermite Curves13.4 Bézier Curves13.4.1 The de Casteljau Algorithm13.4.2 The Bernstein Basis13.4.3 Bézier Derivatives and Their Relationship to the Hermite Form13.5 Subdivision13.5.1 Subdividing Curves in Monomial Form13.5.2 Subdividing Curves in Bézier Form13.6 Splines13.6.1 Rules of the Game13.6.2 Knots13.7 Hermite and Bézier Splines13.8 Continuity13.8.1 Parametric Continuity13.8.2 Geometric Continuity13.8.3 How Smooth Can a Curve Be?13.9 Automatic Tangent Control13.9.1 Catmull-Rom Splines13.9.2 TCB Splines13.9.3 Endpoint Conditions13.10 ExercisesFigure 13.1Figure 13.2Figure 13.3Figure 13.4Figure 13.5Figure 13.6Figure 13.7Figure 13.8Figure 13.9Figure 13.10Figure 13.11Figure 13.12Figure 13.13Figure 13.14Figure 13.15Figure 13.16Figure 13.17Figure 13.18Figure 13.19Figure 13.20Figure 13.21Figure 13.22Figure 13.23Figure 13.24Figure 13.25Figure 13.26Figure 13.27Listing 13.1
Chapter 14 Afterword
14.1 What Next?14.2 Exercises
Appendix A Geometric Tests
A.1 Closest Point on 2D Implicit LineA.2 Closest Point on a Parametric RayA.3 Closest Point on a PlaneA.4 Closest Point on a Circle or SphereA.5 Closest Point in an AABBA.6 Intersection TestsA.7 Intersection of Two Implicit Lines in 2DA.8 Intersection of Two Rays in 3DA.9 Intersection of a Ray and PlaneA.10 Intersection of an AABB and PlaneA.11 Intersection of Three PlanesA.12 Intersection of Ray and a Circle or SphereA.13 Intersection of Two Circles or SpheresA.14 Intersection of a Sphere and AABBA.15 Intersection of a Sphere and a PlaneA.16 Intersection of a Ray and a TriangleA.17 Intersection of Two AABBsA.18 Intersection of a Ray and an AABBFigure A.1Figure A.2Figure A.3Figure A.4Figure A.5Figure A.6Figure A.7Figure A.8Figure A.9Figure A.10Figure A.11Figure A.12Figure A.13Figure A.14Figure A.15Figure A.16Listing A.1Listing A.2Listing A.3Listing A.4Listing A.5Listing A.6
Appendix B Answers to the Exercises
B.1 Chapter 1B.2 Chapter 2B.3 Chapter 3B.4 Chapter 4B.5 Chapter 5B.6 Chapter 6B.7 Chapter 7B.8 Chapter 8B.9 Chapter 9B.10 Chapter 10B.11 Chapter 11B.12 Chapter 12B.13 Chapter 13B.14 Chapter 14
Bibliography

Content preview from 3D Math Primer for Graphics and Game Development, 2nd Edition

Polar Coordinate Systems

First of all, we must note that the universe is spherical.

— Nicolaus Copernicus (1473–1543)

The Cartesian coordinate system isn’t the only system for mapping out space and defining locations precisely. An alternative to the Cartesian system is the polar coordinate system, which is the subject of this chapter. If you’re not very familiar with polar coordinates, it might seem like an esoteric or advanced topic (especially because of the trig), and you might be tempted to gloss over. Please don’t make this mistake. There are many very practical problems in areas such as AI and camera control whose solutions (and inherent difficulties!) can be readily understood in the framework of polar coordinates.

This chapter is organized ...

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3D Math Primer for Graphics and Game Development, 2nd Edition

by Fletcher Dunn, Ian Parberry

Polar Coordinate Systems

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