Hypercomplex Iterations : Distance Estimation and Higher Dimensional Fractals.

By: Dang, YumeiContributor(s): Kauffman, Louis H | Sandin, Daniel JMaterial type: TextTextSeries: Series on Knots and Everything SerPublisher: Singapore : World Scientific Publishing Co Pte Ltd, 1998Copyright date: ©2002Description: 1 online resource (163 pages)Content type: text Media type: computer Carrier type: online resourceISBN: 9789812778604Subject(s): Fractals | Iterative methods (Mathematics) | Mandelbrot sets | QuaternionsGenre/Form: Electronic books.Additional physical formats: Print version:: Hypercomplex Iterations : Distance Estimation and Higher Dimensional FractalsDDC classification: 511.4 LOC classification: QA297.8.D25 2002Online resources: Click to View
Contents:
Intro -- Contents -- Acknowledgements -- Preface -- Part 1 Introduction -- Chapter 1 Hypercomplex Iterations in a Nutshell -- Chapter 2 Deterministic Fractals and Distance Estimation -- 2.1. Fractals and Visualization -- 2.2. Deterministic Fractals Julia Sets and Mandelbrot Sets -- 2.3. Distance Estimation -- Part 2 Classical Analysis: Complex and Quaternionic -- Chapter 3 Distance Estimation in Complex Space -- 3.1. Complex Dynamical Systems -- 3.2. The Quadratic Family Julia Sets and the Mandelbrot Set -- 3.3. The Distance Estimation Formula -- 3.4. Schwarz's Lemma and an Upper Bound of the Distance Estimate -- 3.5. The Koebe 1/4 Theorem and a Lower Bound for the Distance Estimate -- 3.6. An Approximation of the Distance Estimation Formula -- Chapter 4 Quaternion Analysis -- 4.1. The Quaternions -- 4.2. Rotations of 3-Space -- 4.3. Quaternion Polynomials -- 4.4. Quaternion Julia Sets and Mandelbrot Sets -- 4.5. Differential Forms -- 4.6. Regular Functions -- 4.7. Cauchy's Theorem and the Integral Formula -- 4.8. Linear and Quadratic Regular Functions -- 4.9. Difficulties of the Quaternion Analytic Proof of Distance Estimation -- Chapter 5 Quaternions and the Dirac String Trick -- Part 3 Hypercomplex Iterations -- Chapter 6 Quaternion Mandelbrot Sets -- 6.1. Quaternion Mandelbrot Sets -- 6.2. The Distance Estimate for Quaternion Mandelbrot Sets -- Chapter 7 Distance Estimation in Higher Dimensional Spaces -- 7.1. Higher Dimensional Deterministic Fractals -- 7.2. The Cayley Numbers -- 7.3. Distance Estimation in Higher Dimensional Spaces -- 7.4. Calculating the Derivative in Higher Dimensional Space -- 7.5. Another Version of the Distance Estimation Formula -- Part 4 Inverse Iteration Ray Tracing and Virtual Reality -- Chapter 8 Inverse Iteration: An Interactive Visualization -- 8.1. Classical Inverse Iteration -- 8.2. Mappings in the Quaternions.
8.3. The Quaternion Square Root -- 8.4. The nth Roots in Higher Dimensions -- 8.5. Quaternion Julia Sets via Inverse Iteration -- 8.6. Functions Used in the Inverse Iteration Method -- 8.7. An Algorithm for the Inverse Iteration Method -- 8.8. Tree Pruning -- 8.9. Displaying Julia Sets -- Chapter 9 Ray Tracing Methods by Distance Estimation -- 9.1. Distance Estimation via Ray Tracing -- 9.2. A Classical Ray Tracing Algorithm -- 9.3. A Ray Tracing Algorithm Using Distance Estimation -- 9.4. Quaternion Multiplication in the Algorithm -- 9.5. Calculating the Derivative in the Algorithm -- 9.6. Some Important Parameters in the Algorithm -- 9.7. The nth power Family of Quaternion Mandelbrot Sets -- 9.8. The Quadratic Family of Julia Sets -- 9.9. Generalized Quaternion Julia Sets -- 9.10. Disconnected Quaternion Julia Sets -- 9.11. Displaying and Rendering -- 9.11.1. Light models -- 9.11.2. Surface normal -- 9.11.3. Clarity -- 9.11.4. Other Rendering Considerations -- Chapter 10 Quaternion Deterministic Fractals in Virtual Reality -- 10.1. Introduction to Virtual Reality -- 10.2. Parallel Computation -- 10.3. Data Communication -- 10.4. An Improved Display Algorithm -- 10.5. Display of Quaternion Deterministic Fractals in VR -- 10.6. Conclusion -- Appendix A -- Appendix B -- Bibliography -- Index.
Summary: This book is based on the authors' research on rendering images of higher dimensional fractals by a distance estimation technique. It is self-contained, giving a careful treatment of both the known techniques and the authors' new methods. The distance estimation technique was originally applied to Julia sets and the Mandelbrot set in the complex plane. It was justified, through the work of Douady and Hubbard, by deep results in complex analysis. In this book the authors generalise the distance estimation to quaternionic and other higher dimensional fractals, including fractals derived from iteration in the Cayley numbers (octonionic fractals). The generalization is justified by new geometric arguments that circumvent the need for complex analysis. This puts on a firm footing the authors' present work and the second author's earlier work with John Hart and Dan Sandin. The results of this book will be of great interest to mathematicians and computer scientists interested in fractals and computer graphics. Contents: Introduction: Hypercomplex Iteractions in a Nutshell; Deterministic Fractals and Distance Estimation; Classical Analysis: Complex and Quaternionic: Distance Estimation in Complex Space; Quaternion Analysis; Quaternions and the Dirac String Trick; Hypercomplex Iteractions: Quaternion Mandelbrot Sets; Distance Estimation in Higher Dimensional Spaces; Inverse Iteraction, Ray Tracing and Virtual Reality: Inverse Iteraction: An Interactive Visualization; Ray Tracing Methods by Distance Estimation; Quaternion Deterministic Fractals in Virtual Reality. Readership: Mathematicians and computer scientists.
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Intro -- Contents -- Acknowledgements -- Preface -- Part 1 Introduction -- Chapter 1 Hypercomplex Iterations in a Nutshell -- Chapter 2 Deterministic Fractals and Distance Estimation -- 2.1. Fractals and Visualization -- 2.2. Deterministic Fractals Julia Sets and Mandelbrot Sets -- 2.3. Distance Estimation -- Part 2 Classical Analysis: Complex and Quaternionic -- Chapter 3 Distance Estimation in Complex Space -- 3.1. Complex Dynamical Systems -- 3.2. The Quadratic Family Julia Sets and the Mandelbrot Set -- 3.3. The Distance Estimation Formula -- 3.4. Schwarz's Lemma and an Upper Bound of the Distance Estimate -- 3.5. The Koebe 1/4 Theorem and a Lower Bound for the Distance Estimate -- 3.6. An Approximation of the Distance Estimation Formula -- Chapter 4 Quaternion Analysis -- 4.1. The Quaternions -- 4.2. Rotations of 3-Space -- 4.3. Quaternion Polynomials -- 4.4. Quaternion Julia Sets and Mandelbrot Sets -- 4.5. Differential Forms -- 4.6. Regular Functions -- 4.7. Cauchy's Theorem and the Integral Formula -- 4.8. Linear and Quadratic Regular Functions -- 4.9. Difficulties of the Quaternion Analytic Proof of Distance Estimation -- Chapter 5 Quaternions and the Dirac String Trick -- Part 3 Hypercomplex Iterations -- Chapter 6 Quaternion Mandelbrot Sets -- 6.1. Quaternion Mandelbrot Sets -- 6.2. The Distance Estimate for Quaternion Mandelbrot Sets -- Chapter 7 Distance Estimation in Higher Dimensional Spaces -- 7.1. Higher Dimensional Deterministic Fractals -- 7.2. The Cayley Numbers -- 7.3. Distance Estimation in Higher Dimensional Spaces -- 7.4. Calculating the Derivative in Higher Dimensional Space -- 7.5. Another Version of the Distance Estimation Formula -- Part 4 Inverse Iteration Ray Tracing and Virtual Reality -- Chapter 8 Inverse Iteration: An Interactive Visualization -- 8.1. Classical Inverse Iteration -- 8.2. Mappings in the Quaternions.

8.3. The Quaternion Square Root -- 8.4. The nth Roots in Higher Dimensions -- 8.5. Quaternion Julia Sets via Inverse Iteration -- 8.6. Functions Used in the Inverse Iteration Method -- 8.7. An Algorithm for the Inverse Iteration Method -- 8.8. Tree Pruning -- 8.9. Displaying Julia Sets -- Chapter 9 Ray Tracing Methods by Distance Estimation -- 9.1. Distance Estimation via Ray Tracing -- 9.2. A Classical Ray Tracing Algorithm -- 9.3. A Ray Tracing Algorithm Using Distance Estimation -- 9.4. Quaternion Multiplication in the Algorithm -- 9.5. Calculating the Derivative in the Algorithm -- 9.6. Some Important Parameters in the Algorithm -- 9.7. The nth power Family of Quaternion Mandelbrot Sets -- 9.8. The Quadratic Family of Julia Sets -- 9.9. Generalized Quaternion Julia Sets -- 9.10. Disconnected Quaternion Julia Sets -- 9.11. Displaying and Rendering -- 9.11.1. Light models -- 9.11.2. Surface normal -- 9.11.3. Clarity -- 9.11.4. Other Rendering Considerations -- Chapter 10 Quaternion Deterministic Fractals in Virtual Reality -- 10.1. Introduction to Virtual Reality -- 10.2. Parallel Computation -- 10.3. Data Communication -- 10.4. An Improved Display Algorithm -- 10.5. Display of Quaternion Deterministic Fractals in VR -- 10.6. Conclusion -- Appendix A -- Appendix B -- Bibliography -- Index.

This book is based on the authors' research on rendering images of higher dimensional fractals by a distance estimation technique. It is self-contained, giving a careful treatment of both the known techniques and the authors' new methods. The distance estimation technique was originally applied to Julia sets and the Mandelbrot set in the complex plane. It was justified, through the work of Douady and Hubbard, by deep results in complex analysis. In this book the authors generalise the distance estimation to quaternionic and other higher dimensional fractals, including fractals derived from iteration in the Cayley numbers (octonionic fractals). The generalization is justified by new geometric arguments that circumvent the need for complex analysis. This puts on a firm footing the authors' present work and the second author's earlier work with John Hart and Dan Sandin. The results of this book will be of great interest to mathematicians and computer scientists interested in fractals and computer graphics. Contents: Introduction: Hypercomplex Iteractions in a Nutshell; Deterministic Fractals and Distance Estimation; Classical Analysis: Complex and Quaternionic: Distance Estimation in Complex Space; Quaternion Analysis; Quaternions and the Dirac String Trick; Hypercomplex Iteractions: Quaternion Mandelbrot Sets; Distance Estimation in Higher Dimensional Spaces; Inverse Iteraction, Ray Tracing and Virtual Reality: Inverse Iteraction: An Interactive Visualization; Ray Tracing Methods by Distance Estimation; Quaternion Deterministic Fractals in Virtual Reality. Readership: Mathematicians and computer scientists.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.

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