Matrix Transforms for Computer Games and Animation

✍ Scribed by John Vince

• Publisher: Springer Science & Business Media
• Year: 2012
• Tongue: English
• Leaves: 170
• Category: Library

✦ Synopsis

Matrix transforms are ubiquitous within the world of computer graphics, where they have become an invaluable tool in a programmer’s toolkit for solving everything from 2D image scaling to 3D rotation about an arbitrary axis. Virtually every software system and hardware graphics processor uses matrices to undertake operations such as scaling, translation, reflection and rotation. Nevertheless, for some newcomers to the world of computer games and animation, matrix notation can appear obscure and challenging. Matrices and determinants were originally used to solve groups of simultaneous linear equations, and were subsequently embraced by the computer graphics community to describe the geometric operations for manipulating two- and three-dimensional structures. Consequently, to place matrix notation within an historical context, the author provides readers with some useful background to their development, alongside determinants. Although it is assumed that the reader is familiar with everyday algebra and the solution of simultaneous linear equations, Matrix Transforms for Computer Games and Animation does not expect any prior knowledge of matrix notation. It includes chapters on matrix notation, determinants, matrices, 2D transforms, 3D transforms and quaternions, and includes many worked examples to illustrate their practical use.

✦ Table of Contents

Matrix Transforms for Computer Games and Animation
Preface
Contents
Chapter 1: Introduction
1.1 Matrix Transforms
1.2 Mathematics
1.3 The Book's Structure
Chapter 2: Introduction to Matrix Notation
2.1 Introduction
2.2 Solving a Pair of Linear Equations
2.2.1 Graphical Technique
2.2.2 Algebraic Technique
2.2.3 Matrix Technique
2.3 Matrix Multiplication
2.4 Identity Matrix
2.5 Inverse Matrix
2.6 Worked Examples
2.7 Summary
Chapter 3: Determinants
3.1 Introduction
3.2 Linear Equations in Three Unknowns
3.2.1 The Laplace Expansion
3.3 Linear Equations in Four Unknowns
3.4 Worked Examples
3.5 Summary
Chapter 4: Matrices
4.1 Introduction
4.2 Rectangular and Square Matrices
4.3 Matrix Shorthand
4.4 Matrix Addition and Subtraction
4.5 Matrix Scaling
4.6 Matrix Multiplication
4.6.1 Vector Scalar Product
4.6.2 The Vector Product
4.7 The Zero Matrix
4.8 The Negative Matrix
4.9 Diagonal Matrix
4.10 The Identity Matrix
4.11 The Transposed Matrix
4.12 Trace
4.13 Symmetric Matrix
4.14 Antisymmetric Matrix
4.15 Inverse Matrix
4.15.1 Cofactor Matrix
4.16 Orthogonal Matrix
4.17 Worked Examples
4.18 Summary
Chapter 5: 2D Matrix Transforms
5.1 Introduction
5.2 Transforms
5.2.1 Homogeneous Coordinates
5.3 Translation
5.4 Scaling
5.5 Reflection
5.5.1 Reflection About the x and y Axis
5.5.2 Reflection About a Horizontal or Vertical Axis
5.5.3 Reflection in a Line Intersecting the Origin
5.6 Shearing
5.7 Rotation
5.7.1 Rotation About an Arbitrary Point
5.7.2 Rotation and Translation
5.7.3 Composite Rotations
5.8 Change of Axes
5.9 Eigenvectors and Eigenvalues
5.10 Worked Examples
5.11 Summary
Chapter 6: 3D Transforms
6.1 Introduction
6.2 Scaling
6.3 Translation
6.4 Shearing
6.5 Reflection in a Plane Intersecting the Origin
6.6 Rotation
6.6.1 Rotation About an Off-Set Axis
6.6.2 Composite Rotations
6.7 3D Eigenvectors
6.8 Gimbal Lock
6.9 Yaw, Pitch and Roll
6.10 Rotation About an Arbitrary Axis
6.10.1 Matrices
6.10.2 Vectors
6.11 Worked Examples
6.12 Summary
Chapter 7: Quaternions
7.1 Introduction
7.2 Adding and Subtracting Quaternions
7.3 Multiplying Quaternions
7.4 Pure Quaternion
7.5 The Inverse Quaternion
7.6 Unit-Norm Quaternion
7.7 Rotating Points About an Axis
7.8 Roll, Pitch and Yaw Quaternions
7.9 Quaternions in Matrix Form
7.9.1 Vector Method
7.9.2 Matrix Method
7.9.3 Geometric Verification
7.10 Multiple Rotations
7.11 Eigenvalue and Eigenvector
7.12 Rotating About an Off-Set Axis
7.13 Frames of Reference
7.14 Euler Angles to Quaternion
7.15 Worked Examples
7.16 Summary
Chapter 8: Conclusion
Appendix : Composite Point Rotation Sequences
A.1 Euler Rotations
A.2 Rgamma, xRbeta, yRalpha, x
A.3 Rgamma, xRbeta, yRalpha, z
A.4 Rgamma, xRbeta, zRalpha, x
A.5 Rgamma, xRbeta, zRalpha, y
A.6 Rgamma, yRbeta, xRalpha, y
A.7 Rgamma, yRbeta, xRalpha, z
A.8 Rgamma, yRbeta, zRalpha, x
A.9 Rgamma, yRbeta, zRalpha, y
A.10 Rgamma, zRbeta, xRalpha, y
A.11 Rgamma, zRbeta, xRalpha, z
A.12 Rgamma, zRbeta, yRalpha, x
A.13 Rgamma, zRbeta, yRalpha, z
Index