Contents
Introduction
1Cartesian Coordinate Systems
1.11D Mathematics
1.22D Cartesian Space
1.2.1An Example: The Hypothetical City of Cartesia
1.2.2Arbitrary 2D Coordinate Spaces
1.2.3Specifying Locations in 2D Using Cartesian Coordinates
1.33D Cartesian Space
1.3.1Extra Dimension, Extra Axis
1.3.2Specifying Locations in 3D
1.3.3Left-handed versus Right-handed Coordinate Spaces
1.3.4Some Important Conventions Used in This Book
1.4Odds and Ends
1.4.1Summation and Product Notation
1.4.2Interval Notation
1.4.3Angles, Degrees, and Radians
1.4.4Trig Functions
1.4.5Trig Identities
1.5Exercises
2Vectors
2.1Mathematical Definition of Vector,
and Other Boring Stuff
2.2Geometric Definition of Vector
2.3Specifying Vectors with Cartesian Coordinates
2.3.1Vector as a Sequence of Displacements
2.3.2The Zero Vector
2.4Vectors versus Points
2.4.1Relative Position
2.4.2The Relationship between Points and Vectors
2.4.3It's All Relative
2.5Negating a Vector
2.5.1Official Linear Algebra Rules
2.5.2Geometric Interpretation
2.6Vector Multiplication by a Scalar
2.6.1Official Linear Algebra Rules
2.6.2Geometric Interpretation
2.7Vector Addition and Subtraction
2.7.1Official Linear Algebra Rules
2.7.2Geometric Interpretation
2.7.3Displacement Vector from One Point to Another
2.8Vector Magnitude (Length)
2.8.1Official Linear Algebra Rules
2.8.2Geometric Interpretation
2.9Unit Vectors
2.9.1Official Linear Algebra Rules
2.9.2Geometric Interpretation
2.10The Distance Formula
2.11Vector Dot Product
2.11.1Official Linear Algebra Rules
2.11.2Geometric Interpretation
2.12Vector Cross Product
2.12.1Official Linear Algebra Rules
2.12.2Geometric Interpretation
2.13Linear Algebra Identities
2.14Exercises
3Multiple Coordinate Spaces
3.1Why Bother with Multiple Coordinate Spaces?
3.2Some Useful Coordinate Spaces
3.2.1World Space
3.2.2Object Space
3.2.3Camera Space
3.2.4Upright Space
3.3Basis Vectors and Coordinate Space
Transformations
3.3.1Dual Perspectives
3.3.2Specifying Coordinate Spaces
3.3.3Basis Vectors
3.4Nested Coordinate Spaces
3.5In Defense of Upright Space
3.6Exercises
4Introduction to Matrices
4.1Mathematical Definition of Matrix
4.1.1Matrix Dimensions and Notation
4.1.2Square Matrices
4.1.3Vectors as Matrices
4.1.4Matrix Transposition
4.1.5Multiplying a Matrix with a Scalar
4.1.6Multiplying Two Matrices
4.1.7Multiplying a Vector and a Matrix
4.1.8Row versus Column Vectors
4.2Geometric Interpretation of Matrix
4.3The Bigger Picture of Linear Algebra
4.4Exercises
5Matrices and Linear Transformations
5.1Rotation
5.1.1Rotation in 2D
5.1.23D Rotation about Cardinal Axes
5.1.33D Rotation about an Arbitrary Axis
5.2Scale
5.2.1Scaling along the Cardinal Axes
5.2.2Scaling in an Arbitrary Direction
5.3Orthographic Projection
5.3.1Projecting onto a Cardinal Axis or Plane
5.3.2Projecting onto an Arbitrary Line or Plane
5.4Reflection
5.5Shearing
5.6Combining Transformations
5.7Classes of Transformations
5.7.1Linear Transformations
5.7.2Affine Transformations
5.7.3Invertible Transformations
5.7.4Angle-Preserving Transformations
5.7.5Orthogonal Transformations
5.7.6Rigid Body Transformations
5.7.7Summary of Types of Transformations
5.8Exercises
6More on Matrices
6.1Determinant of a Matrix
6.1.1Determinants of 2×2 and 3×3 matrices
6.1.2Minors and Cofactors
6.1.3Determinants of Arbitrary n×n Matrices
6.1.4Geometric Interpretation of Determinant
6.2Inverse of a Matrix
6.2.1The Classical Adjoint
6.2.2Matrix Inverse—Official Linear Algebra Rules
6.2.3Matrix Inverse—Geometric Interpretation
6.3Orthogonal Matrices
6.3.1Orthogonal Matrices—Official Linear Algebra Rules
6.3.2Orthogonal Matrices—Geometric Interpretation
6.3.3Orthogonalizing a Matrix
6.4 4×4 Homogeneous Matrices
6.4.14D Homogeneous Space
6.4.2 4×4 Translation Matrices
6.4.3General Affine Transformations
6.5 4×4 Matrices and Perspective Projection
6.5.1A Pinhole Camera
6.5.2Perspective Projection Matrices
6.6Exercises
7Polar Coordinate Systems
7.12D Polar Space
7.1.1Locating Points by Using 2D Polar Coordinates
7.1.2Aliasing
7.1.3Converting between Cartesian and
Polar Coordinates in 2D
7.2Why Would Anybody Use Polar Coordinates?
7.33D Polar Space
7.3.1Cylindrical Coordinates
7.3.2Spherical Coordinates
7.3.3Some Polar Conventions Useful in 3D Virtual Worlds
7.3.4Aliasing of Spherical Coordinates
7.3.5Converting between Spherical and Cartesian Coordinates
7.4Using Polar Coordinates to Specify Vectors
7.5Exercises
8Rotation in Three Dimensions
8.1What Exactly is “Orientation”?
8.2Matrix Form
8.2.1Which Matrix?
8.2.2Direction Cosines Matrix
8.2.3Advantages of Matrix Form
8.2.4Disadvantages of Matrix Form
8.2.5Summary of Matrix Form
8.3Euler Angles
8.3.1What Are Euler Angles?
8.3.2Other Euler Angle Conventions
8.3.3Advantages of Euler Angles
8.3.4Disadvantages of Euler Angles
8.3.5Summary of Euler Angles
8.4Axis-Angle and Exponential Map
Representations
8.5Quaternions
8.5.1Quaternion Notation
8.5.2What Do Those Four Numbers Mean?
8.5.3Quaternion Negation
8.5.4Identity Quaternion(s)
8.5.5Quaternion Magnitude
8.5.6Quaternion Conjugate and Inverse
8.5.7Quaternion Multiplication
8.5.8Quaternion “Difference”
8.5.9Quaternion Dot Product
8.5.10Quaternion log, exp, and Multiplication by a Scalar
8.5.11Quaternion Exponentiation
8.5.12Quaternion Interpolation, a.k.a. Slerp
8.5.13Advantages and Disadvantages of Quaternions
8.5.14Quaternions as Complex Numbers
8.5.15Summary of Quaternions
8.6Comparison of Methods
8.7Converting between Representations
8.7.1Converting Euler Angles to a Matrix
8.7.2Converting a Matrix to Euler angles
8.7.3Converting a Quaternion to a Matrix
8.7.4Converting a Matrix to a Quaternion
8.7.5Converting Euler Angles to a Quaternion
8.7.6Converting a Quaternion to Euler Angles
8.8Exercises
9Geometric Primitives
9.1Representation Techniques
9.2Lines and Rays
9.2.1Rays
9.2.2Special 2D Representations of Lines
9.2.3Converting between Representations
9.3Spheres and Circles
9.4Bounding Boxes
9.4.1Representing AABBs
9.4.2Computing AABBs
9.4.3AABBs versus Bounding Spheres
9.4.4Transforming AABBs
9.5Planes
9.5.1The Plane Equation: An Implicit Definition of a Plane
9.5.2Defining a Plane by Using Three Points
9.5.3“Best Fit” Plane for More than Three Points
9.5.4Distance from Point to Plane
9.6Triangles
9.6.1Notation
9.6.2Area of a Triangle
9.6.3Barycentric Space
9.6.4Calculating Barycentric Coordinates
9.6.5Special Points
9.7Polygons
9.7.1Simple versus Complex Polygons
9.7.2Convex versus Concave Polygons
9.7.3Triangulation and Fanning
9.8Exercises
10Mathematical Topics
from 3D Graphics
10.1How Graphics Works
10.1.1The Two Major Approaches to Rendering
10.1.2Describing Surface Properties: The BRDF
10.1.3A Very Brief Introduction to Colorimetry
and Radiometry
10.1.4The Rendering Equation
10.2Viewing in 3D
10.2.1Specifying the Output Window
10.2.2Pixel Aspect Ratio
10.2.3The View Frustum
10.2.4Field of View and Zoom
10.2.5Orthographic Projection
10.3Coordinate Spaces
10.3.1Model, World, and Camera Space
10.3.2Clip Space and the Clip Matrix
10.3.3The Clip Matrix: Preparing for Projection
10.3.4The Clip Matrix: Applying Zoom and
Preparing for Clipping
10.3.5Screen Space
10.3.6Summary of Coordinate Spaces
10.4Polygon Meshes
10.4.1Indexed Triangle Mesh
10.4.2Surface Normals
10.5Texture Mapping
10.6The Standard Local Lighting Model
10.6.1The Standard Lighting Equation: Overview
10.6.2The Specular Component
10.6.3The Diffuse Component
10.6.4The Ambient and Emmissive Components
10.6.5The Lighting Equation: Putting It All Together
10.6.6Limitations of the Standard Model
10.6.7Flat and Gouraud Shading
10.7Light Sources
10.7.1Standard Abstract Light Types
10.7.2Light Attenuation
10.7.3 Doom-style Volumetric Lights
10.7.4Precalculated Lighting
10.8Skeletal Animation
10.9Bump Mapping
10.9.1Tangent Space
10.9.2Calculating Tangent Space Basis Vectors
10.10The Real-Time Graphics Pipeline
10.10.1Buffers
10.10.2Delivering the Geometry
10.10.3Vertex-Level Operations
10.10.4Clipping
10.10.5Backface Culling
10.10.6Rasterization, Shading, and Output
10.11Some HLSL Examples
10.11.1Decal Shading and HLSL Basics
10.11.2Basic Per-Pixel Blinn-Phong Lighting
10.11.3Gouraud Shading
10.11.4Bump Mapping
10.11.5Skinned Mesh
10.12Further Reading
10.13Exercises
11Mechanics 1: Linear Kinematics and Calculus
11.1Overview and Other Expectation-Reducing
Remarks
11.1.1What is Left Out?
11.1.2Some Helpful Lies about Our Universe
11.2Basic Quantities and Units
11.3Average Velocity
11.4Instantaneous Velocity and the Derivative
11.4.1Limit Arguments and the Definition of the Derivative
11.4.2Examples of Derivatives
11.4.3Calculating Derivatives from the Definition
11.4.4Notations for the Derivative
11.4.5A Few Differentiation Rules and Shortcuts
11.4.6Derivatives of Some Special Functions
with Taylor Series
11.4.7The Chain Rule
11.5Acceleration
11.6Motion under Constant Acceleration
11.7The Integral
11.7.1Examples of Integrals
11.7.2The Relationship between the Derivative
and the Integral
11.7.3Summary of Calculus
11.8Uniform Circular Motion
11.8.1Uniform Circular Motion in the Plane
11.8.2Uniform Circular Motion in Three Dimensions
11.9Exercises
12Mechanics 2: Linear and Rotational Dynamics
12.1Newton's Three Laws
12.1.1Newton's First Two Laws: Force and Mass
12.1.2Inertial Reference Frames
12.1.3Newton's Third Law
12.2Some Simple Force Laws
12.2.1Gravitational Force
12.2.2Frictional Forces
12.2.3Spring Forces
12.3Momentum
12.3.1Conservation of Momentum
12.3.2The Center of Mass
12.4Impulsive Forces and Collisions
12.4.1Perfectly Inelastic Collisions
12.4.2General Collision Response
12.4.3The Dirac Delta
12.5Rotational Dynamics
12.5.1Rotational Kinematics
12.5.22D Rotational Dynamics
12.5.33D Rotational Dynamics
12.5.4Collision Response with Rotations
12.6Real-Time Rigid Body Simulators
12.6.1Physics Engine State Variables
12.6.2HighLevel Overview
12.6.3Euler Integration
12.6.4Integration of Rotation
12.7Suggested Reading
12.8Exercises
13Curves in 3D
13.1Parametric Polynomial Curves
13.1.1Parametric Curves
13.1.2Polynomial Curves
13.1.3Matrix Notation
13.1.4Two Trivial Types of Curves
13.1.5Endpoints in Monomial Form
13.1.6Velocities and Tangents
13.2Polynomial Interpolation
13.2.1Aitken's Algorithm
13.2.2Lagrange Basis Polynomials
13.2.3Polynomial Interpolation Summary
13.3Hermite Curves
13.4Bézier Curves
13.4.1The de Casteljau Algorithm
13.4.2The Bernstein Basis
13.4.3Bézier Derivatives and Their Relationship
to the Hermite Form
13.5Subdivision
13.5.1Subdividing Curves in Monomial Form
13.5.2Subdividing Curves in Bézier Form
13.6Splines
13.6.1Rules of the Game
13.6.2Knots
13.7Hermite and Bézier Splines
13.8Continuity
13.8.1Parametric Continuity
13.8.2Geometric Continuity
13.8.3How Smooth Can a Curve Be?
13.9Automatic Tangent Control
13.9.1Catmull-Rom Splines
13.9.2TCB Splines
13.9.3Endpoint Conditions
13.10Exercises
14Afterword
14.1What Next?
14.2Exercises
AGeometric Tests
A.1Closest Point on 2D Implicit Line
A.2Closest Point on a Parametric Ray
A.3Closest Point on a Plane
A.4Closest Point on a Circle or Sphere
A.5Closest Point in an AABB
A.6Intersection Tests
A.7Intersection of Two Implicit Lines in 2D
A.8Intersection of Two Rays in 3D
A.9Intersection of a Ray and Plane
A.10Intersection of an AABB and Plane
A.11Intersection of Three Planes
A.12Intersection of Ray and a Circle or Sphere
A.13Intersection of Two Circles or Spheres
A.14Intersection of a Sphere and AABB
A.15Intersection of a Sphere and a Plane
A.16Intersection of a Ray and a Triangle
A.17Intersection of Two AABBs
A.18Intersection of a Ray and an AABB
BAnswers to the Exercises
B.1Chapter 1
B.2Chapter 2
B.3Chapter 3
B.4Chapter 4
B.5Chapter 5
B.6Chapter 6
B.7Chapter 7
B.8Chapter 8
B.9Chapter 9
B.10Chapter 10
B.11Chapter 11
B.12Chapter 12
B.13Chapter 13
B.14Chapter 14
So much time, and so little to do!
Strike that, reverse it.
— Willy Wonka