Quaternions for Computer Graphics

Sir William Rowan Hamilton was a genius, and will be remembered for his significant contributions to physics and mathematics. The Hamiltonian, which is used in quantum physics to describe the total energy of a system, would have been a major achievement for anyone, but Hamilton also invented quaternions, which paved the way for modern vector analysis. Quaternions are one of the most documented inventions in the history of mathematics, and this book is about their invention, and how they are used to rotate vectors about an arbitrary axis. Apart from introducing the reader to the features of quaternions and their associated algebra, the book provides valuable historical facts that bring the subject alive. Quaternions for Computer Graphics introduces the reader to quaternion algebra by describing concepts of sets, groups, fields and rings. It also includes chapters on imaginary quantities, complex numbers and the complex plane, which are essential to understanding quaternions. The book contains many illustrations and worked examples, which make it essential reading for students, academics, researchers and professional practitioners. Quaternions for Computer Graphics 3 Preface 6 Contents 9 Chapter 1: Introduction 13 1.1 Rotation Transforms 13 1.2 The Reader 13 1.3 Aims and Objectives of This Book 13 1.4 Mathematical Techniques 14 1.5 Assumptions Made in This Book 14 Chapter 2: Number Sets and Algebra 15 2.1 Introduction 15 2.2 Number Sets 15 2.2.1 Natural Numbers 15 2.2.2 Real Numbers 15 2.2.3 Integers 16 2.2.4 Rational Numbers 16 2.3 Arithmetic Operations 16 2.4 Axioms 17 2.5 Expressions 18 2.6 Equations 19 2.7 Ordered Pairs 19 2.8 Groups, Rings and Fields 20 2.8.1 Groups 20 2.8.2 Abelian Group 22 2.8.3 Rings 22 2.8.4 Fields 22 2.8.5 Division Ring 23 2.9 Summary 23 2.9.1 Summary of Definitions 23 Chapter 3: Complex Numbers 25 3.1 Introduction 25 3.2 Imaginary Numbers 25 3.3 Powers of i 25 3.4 Complex Numbers 27 3.5 Adding and Subtracting Complex Numbers 28 3.6 Multiplying a Complex Number by a Scalar 28 3.7 Complex Number Products 28 3.7.1 Square of a Complex Number 29 3.8 Norm of a Complex Number 29 3.9 Complex Conjugate 30 3.10 Quotient of Two Complex Numbers 30 3.11 Inverse of a Complex Number 31 3.12 Square-Root of i 31 3.13 Field Structure 33 3.14 Ordered Pairs 33 3.14.1 Multiplying by a Scalar 34 3.14.2 Complex Conjugate 35 3.14.3 Quotient 35 3.14.4 Inverse 35 3.15 Matrix Representation of a Complex Number 36 3.15.1 Adding and Subtracting 37 3.15.2 The Product 37 3.15.3 The Square of the Norm 37 3.15.4 The Complex Conjugate 37 3.15.5 The Inverse 38 3.15.6 Quotient 38 3.16 Summary 39 3.16.1 Summary of Operations 39 3.17 Worked Examples 41 Chapter 4: The Complex Plane 44 4.1 Introduction 44 4.2 Some History 44 4.3 The Complex Plane 45 4.4 Polar Representation 48 4.5 Rotors 52 4.6 Summary 53 4.6.1 Summary of Operations 53 4.7 Worked Examples 54 Chapter 5: Quaternion Algebra 57 5.1 Introduction 57 5.2 Some History 59 5.3 Defining a Quaternion 63 5.3.1 The Quaternion Units 65 5.3.2 Example of Quaternion Products 66 5.4 Algebraic Definition 66 5.5 Adding and Subtracting Quaternions 67 5.6 Real Quaternion 67 5.7 Multiplying a Quaternion by a Scalar 67 5.8 Pure Quaternion 68 5.9 Unit Quaternion 69 5.10 Additive Form of a Quaternion 70 5.11 Binary Form of a Quaternion 70 5.12 The Conjugate 71 5.13 Norm of a Quaternion 72 5.14 Normalised Quaternion 72 5.15 Quaternion Products 73 5.15.1 Product of Pure Quaternions 73 5.15.2 Product of Two Unit-Norm Quaternions 73 5.15.3 Square of a Quaternion 74 5.15.4 Norm of the Quaternion Product 75 5.16 Inverse Quaternion 75 5.17 Matrices 77 5.17.1 Orthogonal Matrix 77 5.18 Quaternion Algebra 78 5.19 Summary 78 5.19.1 Summary of Operations 79 5.20 Worked Examples 80 Chapter 6: 3D Rotation Transforms 82 6.1 Introduction 82 6.2 3D Rotation Transforms 82 6.3 Rotating About a Cartesian Axis 82 6.4 Rotate About an Off-Set Axis 83 6.5 Composite Rotations 84 6.6 Rotating About an Arbitrary Axis 87 6.6.1 Matrices 87 6.6.2 Vectors 89 6.7 Summary 92 6.7.1 Summary of Transforms 92 6.8 Worked Examples 93 Chapter 7: Quaternions in Space 98 7.1 Introduction 98 7.2 Some History 99 7.2.1 Composition Algebras 100 7.3 Quaternion Products 102 7.3.1 Special Case 102 7.3.2 General Case 106 7.4 Quaternions in Matrix Form 110 7.4.1 Vector Method 110 7.4.2 Matrix Method 112 7.4.3 Geometric Verification 115 7.5 Multiple Rotations 117 7.6 Eigenvalue and Eigenvector 119 7.7 Rotating About an Off-Set Axis 120 7.8 Frames of Reference 122 7.9 Interpolating Quaternions 123 7.10 Converting a Rotation Matrix to a Quaternion 127 7.11 Euler Angles to Quaternion 129 7.12 Summary 132 7.12.1 Summary of Operations 132 7.13 Worked Examples 134 Chapter 8: Conclusion 139 Appendix : Eigenvectors and Eigenvalues 140 References 144 Index 146