3D Math Primer for Graphics and Game Development · 2nd Edition by Dunn, Fletcher -- Read -- Imperial Library of Trantor

Index

Preliminaries Dedication Epigraph Acknowledgments Introduction

Who Should Read This Book Why You Should Read This Book What You Should Know before Reading This Book Overview Find a Bug in This Book?

Chapter 1 Cartesian Coordinate Systems

1.1 1D Mathematics 1.2 2D Cartesian Space

1.2.1 An Example: The Hypothetical City of Cartesia 1.2.2 Arbitrary 2D Coordinate Spaces 1.2.3 Specifying Locations in 2D Using Cartesian Coordinates

1.3 3D Cartesian Space

1.3.1 Extra Dimension, Extra Axis 1.3.2 Specifying Locations in 3D 1.3.3 Left-handed versus Right-handed Coordinate Spaces 1.3.4 Some Important Conventions Used in This Book

1.4 Odds and Ends

1.4.1 Summation and Product Notation 1.4.2 Interval Notation 1.4.3 Angles, Degrees, and Radians 1.4.4 Trig Functions 1.4.5 Trig Identities

1.5 Exercises Figure 1.1 Figure 1.2 Figure 1.3 Figure 1.4 Figure 1.5 Figure 1.6 Figure 1.7 Figure 1.8 Figure 1.9 Figure 1.10 Figure 1.11 Figure 1.12 Figure 1.13 Figure 1.14 Figure 1.15 Figure 1.16 Figure 1.17 Figure 1.18 Figure 1.19 Figure 1.20 Figure 1.21 Table 1.1 Table 1.2

Chapter 2 Vectors

2.1 Mathematical Definition of Vector, and Other Boring Stuff 2.2 Geometric Definition of Vector 2.3 Specifying Vectors with Cartesian Coordinates

2.3.1 Vector as a Sequence of Displacements 2.3.2 The Zero Vector

2.4 Vectors versus Points

2.4.1 Relative Position 2.4.2 The Relationship between Points and Vectors 2.4.3 It’s All Relative

2.5 Negating a Vector

2.5.1 Official Linear Algebra Rules 2.5.2 Geometric Interpretation

2.6 Vector Multiplication by a Scalar

2.6.1 Official Linear Algebra Rules 2.6.2 Geometric Interpretation

2.7 Vector Addition and Subtraction

2.7.1 Official Linear Algebra Rules 2.7.2 Geometric Interpretation 2.7.3 Displacement Vector from One Point to Another

2.8 Vector Magnitude (Length)

2.8.1 Official Linear Algebra Rules 2.8.2 Geometric Interpretation

2.9 Unit Vectors

2.9.1 Official Linear Algebra Rules 2.9.2 Geometric Interpretation

2.10 The Distance Formula 2.11 Vector Dot Product

2.11.1 Official Linear Algebra Rules 2.11.2 Geometric Interpretation

2.12 Vector Cross Product

2.12.1 Official Linear Algebra Rules 2.12.2 Geometric Interpretation

2.13 Linear Algebra Identities 2.14 Exercises Figure 2.1 Figure 2.2 Figure 2.3 Figure 2.4 Figure 2.5 Figure 2.6 Figure 2.7 Figure 2.8 Figure 2.9 Figure 2.10 Figure 2.11 Figure 2.12 Figure 2.13 Figure 2.14 Figure 2.15 Figure 2.16 Figure 2.17 Figure 2.18 Figure 2.19 Figure 2.20 Figure 2.21 Figure 2.22 Figure 2.23 Figure 2.24 Figure 2.25 Figure 2.26 Figure 2.27 Figure 2.28 Figure 2.29 Figure 2.30 Table 2.1 Table 2.2

Chapter 3 Multiple Coordinate Spaces

3.1 Why Bother with Multiple Coordinate Spaces? 3.2 Some Useful Coordinate Spaces

3.2.1 World Space 3.2.2 Object Space 3.2.3 Camera Space 3.2.4 Upright Space

3.3 Basis Vectors and Coordinate Space Transformations

3.3.1 Dual Perspectives 3.3.2 Specifying Coordinate Spaces 3.3.3 Basis Vectors

3.4 Nested Coordinate Spaces 3.5 In Defense of Upright Space 3.6 Exercises Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Figure 3.7 Figure 3.8 Figure 3.9 Figure 3.10 Figure 3.11 Figure 3.12 Figure 3.13 Table 3.1

Chapter 4 Introduction to Matrices

4.1 Mathematical Definition of Matrix

4.1.1 Matrix Dimensions and Notation 4.1.2 Square Matrices 4.1.3 Vectors as Matrices 4.1.4 Matrix Transposition 4.1.5 Multiplying a Matrix with a Scalar 4.1.6 Multiplying Two Matrices 4.1.7 Multiplying a Vector and a Matrix 4.1.8 Row versus Column Vectors

4.2 Geometric Interpretation of Matrix 4.3 The Bigger Picture of Linear Algebra 4.4 Exercises Figure 4.1 Figure 4.2 Figure 4.3 Figure 4.4 Figure 4.5

Chapter 5 Matrices and Linear Transformations

5.1 Rotation

5.1.1 Rotation in 2D 5.1.2 3D Rotation about Cardinal Axes 5.1.3 3D Rotation about an Arbitrary Axis

5.2 Scale

5.2.1 Scaling along the Cardinal Axes 5.2.2 Scaling in an Arbitrary Direction

5.3 Orthographic Projection

5.3.1 Projecting onto a Cardinal Axis or Plane 5.3.2 Projecting onto an Arbitrary Line or Plane

5.4 Reflection 5.5 Shearing 5.6 Combining Transformations 5.7 Classes of Transformations

5.7.1 Linear Transformations 5.7.2 Affine Transformations 5.7.3 Invertible Transformations 5.7.4 Angle-Preserving Transformations 5.7.5 Orthogonal Transformations 5.7.6 Rigid Body Transformations 5.7.7 Summary of Types of Transformations

5.8 Exercises Figure 5.1 Figure 5.2 Figure 5.3 Figure 5.4 Figure 5.5 Figure 5.6 Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Table 5.1

Chapter 6 More on Matrices

6.1 Determinant of a Matrix

6.1.1 Determinants of 2 × 2 and 3 × 3 matrices 6.1.2 Minors and Cofactors 6.1.3 Determinants of Arbitrary n × × n Matrices 6.1.4 Geometric Interpretation of Determinant

6.2 Inverse of a Matrix

6.2.1 The Classical Adjoint 6.2.2 Matrix Inverse—Official Linear Algebra Rules 6.2.3 Matrix Inverse—Geometric Interpretation

6.3 Orthogonal Matrices

6.3.1 Orthogonal Matrices—Official Linear Algebra Rules 6.3.2 Orthogonal Matrices—Geometric Interpretation 6.3.3 Orthogonalizing a Matrix

6.4 4 × 4 Homogeneous Matrices

6.4.1 4D Homogeneous Space 6.4.2 4 × 4 Translation Matrices 6.4.3 General Affine Transformations

6.5 4 × 4 Matrices and Perspective Projection

6.5.1 A Pinhole Camera 6.5.2 Perspective Projection Matrices

6.6 Exercises Figure 6.1 Figure 6.2 Figure 6.3 Figure 6.4 Figure 6.5 Figure 6.6 Figure 6.7 Figure 6.8 Figure 6.9

Chapter 7 Polar Coordinate Systems

7.1 2D Polar Space

7.1.1 Locating Points by Using 2D Polar Coordinates 7.1.2 Aliasing 7.1.3 Converting between Cartesian and Polar Coordinates in 2D

7.2 Why Would Anybody Use Polar Coordinates? 7.3 3D Polar Space

7.3.1 Cylindrical Coordinates 7.3.2 Spherical Coordinates 7.3.3 Some Polar Conventions Useful in 3D Virtual Worlds 7.3.4 Aliasing of Spherical Coordinates 7.3.5 Converting between Spherical and Cartesian Coordinates

7.4 Using Polar Coordinates to Specify Vectors 7.5 Exercises Figure 7.1 Figure 7.2 Figure 7.3 Figure 7.4 Figure 7.5 Figure 7.6 Figure 7.7 Figure 7.8 Listing 7.1 Listing 7.2 Listing 7.3 Listing 7.4

Chapter 8 Rotation in Three Dimensions

8.1 What Exactly is “Orientation”? 8.2 Matrix Form

8.2.1 Which Matrix? 8.2.2 Direction Cosines Matrix 8.2.3 Advantages of Matrix Form 8.2.4 Disadvantages of Matrix Form 8.2.5 Summary of Matrix Form

8.3 Euler Angles

8.3.1 What Are Euler Angles? 8.3.2 Other Euler Angle Conventions 8.3.3 Advantages of Euler Angles 8.3.4 Disadvantages of Euler Angles 8.3.5 Summary of Euler Angles

8.4 Axis-Angle and Exponential Map Representations 8.5 Quaternions

8.5.1 Quaternion Notation 8.5.2 What Do Those Four Numbers Mean? 8.5.3 Quaternion Negation 8.5.4 Identity Quaternion(s) 8.5.5 Quaternion Magnitude 8.5.6 Quaternion Conjugate and Inverse 8.5.7 Quaternion Multiplication 8.5.8 Quaternion “Difference” 8.5.9 Quaternion Dot Product 8.5.10 Quaternion log, exp, and Multiplication by a Scalar 8.5.11 Quaternion Exponentiation 8.5.12 Quaternion Interpolation, a.k.a. Slerp 8.5.13 Advantages and Disadvantages of Quaternions 8.5.14 Quaternions as Complex Numbers 8.5.15 Summary of Quaternions

8.6 Comparison of Methods 8.7 Converting between Representations

8.7.1 Converting Euler Angles to a Matrix 8.7.2 Converting a Matrix to Euler angles 8.7.3 Converting a Quaternion to a Matrix 8.7.4 Converting a Matrix to a Quaternion 8.7.5 Converting Euler Angles to a Quaternion 8.7.6 Converting a Quaternion to Euler Angles

8.8 Exercises Figure 8.1 Figure 8.2 Figure 8.3 Figure 8.4 Figure 8.5 Figure 8.6 Figure 8.7 Figure 8.8 Figure 8.9 Figure 8.10 Figure 8.11 Figure 8.12 Figure 8.13 Table 8.1 Table 8.2 Listing 8.1 Listing 8.2 Listing 8.3 Listing 8.4 Listing 8.5 Listing 8.6 Listing 8.7

Chapter 9 Geometric Primitives

9.1 Representation Techniques 9.2 Lines and Rays

9.2.1 Rays 9.2.2 Special 2D Representations of Lines 9.2.3 Converting between Representations

9.3 Spheres and Circles 9.4 Bounding Boxes

9.4.1 Representing AABBs 9.4.2 Computing AABBs 9.4.3 AABBs versus Bounding Spheres 9.4.4 Transforming AABBs

9.5 Planes

9.5.1 The Plane Equation: An Implicit Definition of a Plane 9.5.2 Defining a Plane by Using Three Points 9.5.3 “Best Fit” Plane for More than Three Points 9.5.4 Distance from Point to Plane

9.6 Triangles

9.6.1 Notation 9.6.2 Area of a Triangle 9.6.3 Barycentric Space 9.6.4 Calculating Barycentric Coordinates 9.6.5 Special Points

9.7 Polygons

9.7.1 Simple versus Complex Polygons 9.7.2 Convex versus Concave Polygons 9.7.3 Triangulation and Fanning

9.8 Exercises Figure 9.1 Figure 9.2 Figure 9.3 Figure 9.4 Figure 9.5 Figure 9.6 Figure 9.7 Figure 9.8 Figure 9.9 Figure 9.10 Figure 9.11 Figure 9.12 Figure 9.13 Figure 9.14 Figure 9.15 Figure 9.16 Figure 9.17 Figure 9.18 Figure 9.19 Figure 9.20 Figure 9.21 Figure 9.22 Figure 9.23 Figure 9.24 Figure 9.25 Figure 9.26 Figure 9.27 Figure 9.28 Figure 9.29 Figure 9.30 Listing 9.1 Listing 9.2 Listing 9.3 Listing 9.4 Listing 9.5 Listing 9.6 Listing 9.7

Chapter 10 Mathematical Topics from 3D Graphics

10.1 How Graphics Works

10.1.1 The Two Major Approaches to Rendering 10.1.2 Describing Surface Properties: The BRDF 10.1.3 A Very Brief Introduction to Colorimetry and Radiometry 10.1.4 The Rendering Equation

10.2 Viewing in 3D

10.2.1 Specifying the Output Window 10.2.2 Pixel Aspect Ratio 10.2.3 The View Frustum 10.2.4 Field of View and Zoom 10.2.5 Orthographic Projection

10.3 Coordinate Spaces

10.3.1 Model, World, and Camera Space 10.3.2 Clip Space and the Clip Matrix 10.3.3 The Clip Matrix: Preparing for Projection 10.3.4 The Clip Matrix: Applying Zoom and Preparing for Clipping 10.3.5 Screen Space 10.3.6 Summary of Coordinate Spaces

10.4 Polygon Meshes

10.4.1 Indexed Triangle Mesh 10.4.2 Surface Normals

10.5 Texture Mapping 10.6 The Standard Local Lighting Model

10.6.1 The Standard Lighting Equation: Overview 10.6.2 The Specular Component 10.6.3 The Diffuse Component 10.6.4 The Ambient and Emmissive Components 10.6.5 The Lighting Equation: Putting It All Together 10.6.6 Limitations of the Standard Model 10.6.7 Flat and Gouraud Shading

10.7 Light Sources

10.7.1 Standard Abstract Light Types 10.7.2 Light Attenuation 10.7.3 Doom-style Volumetric Lights 10.7.4 Precalculated Lighting

10.8 Skeletal Animation 10.9 Bump Mapping

10.9.1 Tangent Space 10.9.2 Calculating Tangent Space Basis Vectors

10.10 The Real-Time Graphics Pipeline

10.10.1 Buffers 10.10.2 Delivering the Geometry 10.10.3 Vertex-Level Operations 10.10.4 Clipping 10.10.5 Backface Culling 10.10.6 Rasterization, Shading, and Output

10.11 Some HLSL Examples

10.11.1 Decal Shading and HLSL Basics 10.11.2 Basic Per-Pixel Blinn-Phong Lighting 10.11.3 Gouraud Shading 10.11.4 Bump Mapping 10.11.5 Skinned Mesh

10.12 Further Reading 10.13 Exercises Figure 10.1 Figure 10.2 Figure 10.3 Figure 10.4 Figure 10.5 Figure 10.6 Figure 10.7 Figure 10.8 Figure 10.9 Figure 10.10 Figure 10.11 Figure 10.12 Figure 10.13 Figure 10.14 Figure 10.15 Figure 10.16 Figure 10.17 Figure 10.18 Figure 10.19 Figure 10.20 Figure 10.21 Figure 10.22 Figure 10.23 Figure 10.24 Figure 10.25 Figure 10.26 Figure 10.27 Figure 10.28 Figure 10.29 Figure 10.30 Figure 10.31 Figure 10.32 Figure 10.33 Figure 10.34 Figure 10.35 Figure 10.36 Figure 10.37 Figure 10.38 Figure 10.39 Figure 10.40 Figure 10.41 Figure 10.42 Figure 10.43 Figure 10.44 Table 10.1 Table 10.2 Table 10.3 Table 10.4 Table 10.5 Table 10.6 Listing 10.1 Listing 10.2 Listing 10.3 Listing 10.4 Listing 10.5 Listing 10.6 Listing 10.7 Listing 10.8 Listing 10.9 Listing 10.10 Listing 10.11 Listing 10.12 Listing 10.13 Listing 10.14 Listing 10.15 Listing 10.16 Listing 10.17 Listing 10.18 Listing 10.19 Listing 10.20 Listing 10.21 Listing 10.22 Listing 10.23 Listing 10.24

Chapter 11 Mechanics 1: Linear Kinematics and Calculus

11.1 Overview and Other Expectation-Reducing Remarks

11.1.1 What is Left Out? 11.1.2 Some Helpful Lies about Our Universe

11.2 Basic Quantities and Units 11.3 Average Velocity 11.4 Instantaneous Velocity and the Derivative

11.4.1 Limit Arguments and the Definition of the Derivative 11.4.2 Examples of Derivatives 11.4.3 Calculating Derivatives from the Definition 11.4.4 Notations for the Derivative 11.4.5 A Few Differentiation Rules and Shortcuts 11.4.6 Derivatives of Some Special Functions with Taylor Series 11.4.7 The Chain Rule

11.5 Acceleration 11.6 Motion under Constant Acceleration 11.7 The Integral

11.7.1 Examples of Integrals 11.7.2 The Relationship between the Derivative and the Integral 11.7.3 Summary of Calculus

11.8 Uniform Circular Motion

11.8.1 Uniform Circular Motion in the Plane 11.8.2 Uniform Circular Motion in Three Dimensions

11.9 Exercises Figure 11.1 Figure 11.2 Figure 11.3 Figure 11.4 Figure 11.5 Figure 11.6 Figure 11.7 Figure 11.8 Figure 11.9 Figure 11.10 Figure 11.11 Figure 11.12 Figure 11.13 Figure 11.14 Figure 11.15 Figure 11.16 Figure 11.17 Table 11.1 Table 11.2 Table 11.3

Chapter 12 Mechanics 2: Linear and Rotational Dynamics

12.1 Newton’s Three Laws

12.1.1 Newton’s First Two Laws: Force and Mass 12.1.2 Inertial Reference Frames 12.1.3 Newton’s Third Law

12.2 Some Simple Force Laws

12.2.1 Gravitational Force 12.2.2 Frictional Forces 12.2.3 Spring Forces

12.3 Momentum

12.3.1 Conservation of Momentum 12.3.2 The Center of Mass

12.4 Impulsive Forces and Collisions

12.4.1 Perfectly Inelastic Collisions 12.4.2 General Collision Response 12.4.3 The Dirac Delta

12.5 Rotational Dynamics

12.5.1 Rotational Kinematics 12.5.2 2D Rotational Dynamics 12.5.3 3D Rotational Dynamics 12.5.4 Collision Response with Rotations

12.6 Real-Time Rigid Body Simulators

12.6.1 Physics Engine State Variables 12.6.2 HighLevel Overview 12.6.3 Euler Integration 12.6.4 Integration of Rotation

12.7 Suggested Reading 12.8 Exercises Figure 12.1 Figure 12.2 Figure 12.3 Figure 12.4 Figure 12.5 Figure 12.6 Figure 12.7 Figure 12.8 Figure 12.9 Figure 12.10 Figure 12.11 Figure 12.12 Figure 12.13 Figure 12.14 Figure 12.15 Figure 12.16 Figure 12.17 Figure 12.18 Figure 12.19 Figure 12.20 Figure 12.21 Figure 12.22 Table 12.1 Table 12.2 Listing 12.1 Listing 12.2 Listing 12.3 Listing 12.4 Listing 12.5 Listing 12.6 Listing 12.7

Chapter 13 Curves in 3D

13.1 Parametric Polynomial Curves

13.1.1 Parametric Curves 13.1.2 Polynomial Curves 13.1.3 Matrix Notation 13.1.4 Two Trivial Types of Curves 13.1.5 Endpoints in Monomial Form 13.1.6 Velocities and Tangents

13.2 Polynomial Interpolation

13.2.1 Aitken’s Algorithm 13.2.2 Lagrange Basis Polynomials 13.2.3 Polynomial Interpolation Summary

13.3 Hermite Curves 13.4 Bézier Curves

13.4.1 The de Casteljau Algorithm 13.4.2 The Bernstein Basis 13.4.3 Bézier Derivatives and Their Relationship to the Hermite Form

13.5 Subdivision

13.5.1 Subdividing Curves in Monomial Form 13.5.2 Subdividing Curves in Bézier Form

13.6 Splines

13.6.1 Rules of the Game 13.6.2 Knots

13.7 Hermite and Bézier Splines 13.8 Continuity

13.8.1 Parametric Continuity 13.8.2 Geometric Continuity 13.8.3 How Smooth Can a Curve Be?

13.9 Automatic Tangent Control

13.9.1 Catmull-Rom Splines 13.9.2 TCB Splines 13.9.3 Endpoint Conditions

13.10 Exercises Figure 13.1 Figure 13.2 Figure 13.3 Figure 13.4 Figure 13.5 Figure 13.6 Figure 13.7 Figure 13.8 Figure 13.9 Figure 13.10 Figure 13.11 Figure 13.12 Figure 13.13 Figure 13.14 Figure 13.15 Figure 13.16 Figure 13.17 Figure 13.18 Figure 13.19 Figure 13.20 Figure 13.21 Figure 13.22 Figure 13.23 Figure 13.24 Figure 13.25 Figure 13.26 Figure 13.27 Listing 13.1

Chapter 14 Afterword

14.1 What Next? 14.2 Exercises

Appendix A Geometric Tests

A.1 Closest Point on 2D Implicit Line A.2 Closest Point on a Parametric Ray A.3 Closest Point on a Plane A.4 Closest Point on a Circle or Sphere A.5 Closest Point in an AABB A.6 Intersection Tests A.7 Intersection of Two Implicit Lines in 2D A.8 Intersection of Two Rays in 3D A.9 Intersection of a Ray and Plane A.10 Intersection of an AABB and Plane A.11 Intersection of Three Planes A.12 Intersection of Ray and a Circle or Sphere A.13 Intersection of Two Circles or Spheres A.14 Intersection of a Sphere and AABB A.15 Intersection of a Sphere and a Plane A.16 Intersection of a Ray and a Triangle A.17 Intersection of Two AABBs A.18 Intersection of a Ray and an AABB Figure A.1 Figure A.2 Figure A.3 Figure A.4 Figure A.5 Figure A.6 Figure A.7 Figure A.8 Figure A.9 Figure A.10 Figure A.11 Figure A.12 Figure A.13 Figure A.14 Figure A.15 Figure A.16 Listing A.1 Listing A.2 Listing A.3 Listing A.4 Listing A.5 Listing A.6

Appendix B Answers to the Exercises

B.1 Chapter 1 B.2 Chapter 2 B.3 Chapter 3 B.4 Chapter 4 B.5 Chapter 5 B.6 Chapter 6 B.7 Chapter 7 B.8 Chapter 8 B.9 Chapter 9 B.10 Chapter 10 B.11 Chapter 11 B.12 Chapter 12 B.13 Chapter 13 B.14 Chapter 14

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