Contents 8
Acknowledgements 12
Preface 14
Part 1 Introduction 17
Chapter 1 Hypercomplex Iterations in a Nutshell 19
Chapter 2 Deterministic Fractals and Distance Estimation 27
2.1. Fractals and Visualization 27
2.2. Deterministic Fractals Julia Sets and Mandelbrot Sets 28
2.3. Distance Estimation 29
Part 2 Classical Analysis: Complex and Quaternionic 31
Chapter 3 Distance Estimation in Complex Space 33
3.1. Complex Dynamical Systems 33
3.2. The Quadratic Family Julia Sets and the Mandelbrot Set 34
3.3. The Distance Estimation Formula 37
3.4. Schwarz's Lemma and an Upper Bound of the Distance Estimate 39
3.5. The Koebe 1/4 Theorem and a Lower Bound for the Distance Estimate 45
3.6. An Approximation of the Distance Estimation Formula 51
Chapter 4 Quaternion Analysis 55
4.1. The Quaternions 56
4.2. Rotations of 3-Space 57
4.3. Quaternion Polynomials 58
4.4. Quaternion Julia Sets and Mandelbrot Sets 59
4.5. Differential Forms 60
4.6. Regular Functions 63
4.7. Cauchy's Theorem and the Integral Formula 64
4.8. Linear and Quadratic Regular Functions 69
4.9. Difficulties of the Quaternion Analytic Proof of Distance Estimation 70
Chapter 5 Quaternions and the Dirac String Trick 73
Part 3 Hypercomplex Iterations 79
Chapter 6 Quaternion Mandelbrot Sets 81
6.1. Quaternion Mandelbrot Sets 81
6.2. The Distance Estimate for Quaternion Mandelbrot Sets 81
Chapter 7 Distance Estimation in Higher Dimensional Spaces 87
7.1. Higher Dimensional Deterministic Fractals 87
7.2. The Cayley Numbers 89
7.3. Distance Estimation in Higher Dimensional Spaces 90
7.4. Calculating the Derivative in Higher Dimensional Space 93
7.5. Another Version of the Distance Estimation Formula 98
Part 4 Inverse Iteration Ray Tracing and Virtual Reality 105
Chapter 8 Inverse Iteration: An Interactive Visualization 107
8.1. Classical Inverse Iteration 107
8.2. Mappings in the Quaternions 109
8.3. The Quaternion Square Root 110
8.4. The nth Roots in Higher Dimensions 112
8.5. Quaternion Julia Sets via Inverse Iteration 113
8.6. Functions Used in the Inverse Iteration Method 114
8.7. An Algorithm for the Inverse Iteration Method 116
8.8. Tree Pruning 118
8.9. Displaying Julia Sets 120
Chapter 9 Ray Tracing Methods by Distance Estimation 123
9.1. Distance Estimation via Ray Tracing 123
9.2. A Classical Ray Tracing Algorithm 124
9.3. A Ray Tracing Algorithm Using Distance Estimation 124
9.4. Quaternion Multiplication in the Algorithm 126
9.5. Calculating the Derivative in the Algorithm 127
9.6. Some Important Parameters in the Algorithm 129
9.7. The nth power Family of Quaternion Mandelbrot Sets 130
9.8. The Quadratic Family of Julia Sets 132
9.9. Generalized Quaternion Julia Sets 134
9.10. Disconnected Quaternion Julia Sets 137
9.11. Displaying and Rendering 138
9.11.1. Light models 138
9.11.2. Surface normal 139
9.11.3. Clarity 141
9.11.4. Other Rendering Considerations 143
Chapter 10 Quaternion Deterministic Fractals in Virtual Reality 145
10.1. Introduction to Virtual Reality 145
10.2. Parallel Computation 146
10.3. Data Communication 147
10.4. An Improved Display Algorithm 148
10.5. Display of Quaternion Deterministic Fractals in VR 149
10.6. Conclusion 150
Appendix A 151
Appendix B 153
Bibliography 155
Index 159