Hypercomplex iterations: distance estimation and higher dimensional fractals
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2002
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| Schriftenreihe: | Series on knots and everything
17 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XV, 144 S. graph. Darst. 2 CD-ROMs (12 cm) |
| ISBN: | 9810232969 |
Internformat
MARC
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Datensatz im Suchindex
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| adam_text | Titel: Hypercomplex iterations
Autor: Dang, Yumei
Jahr: 2002
Contents Acknowledgements xi Preface xiii Part 1 Introduction 1 Chapter 1 Hypercomplex Iterations in a Nutshell 3 Chapter 2 Deterministic Fractals and Distance Estimation 11 2.1. Fractals and Visualization................. 11 2.2. Deterministic Fractals, Julia Sets and Mandelbrot Sets....................... 12 2.3. Distance Estimation.................... 13 Part 2 Classical Analysis: Complex and Quaternionic 15 Chapter 3 Distance Estimation in Complex Space 17 3.1. Complex Dynamical Systems ............... 17 3.2. The Quadratic Family, Julia Sets and the Mandelbrot Set....................... 18 3.3. The Distance Estimation Formula............. 21 vii
Contents viii 3.4. Schwarz’s Lemma and an Upper Bound of the Distance Estimate ..................... 23 3.5. The Koebe 1/4 Theorem and a Lower Bound for the Distance Estimate ..................... 29 3.6. An Approximation of the Distance Estimation Formula........................... 35 Chapter 4 Quaternion Analysis 39 4.1. The Quaternions...................... 40 4.2. Rotations of 3-Space.................... 41 4.3. Quaternion Polynomials.................. 42 4.4. Quaternion Julia Sets and Mandelbrot Sets....... 43 4.5. Differential Forms...................... 44 4.6. Regular Functions..................^ . . . 47 4.7. Cauchy’s Theorem and the Integral Formula.......48 4.8. Linear and Quadratic Regular Functions......... 53 4.9. Difficulties of the Quaternion Analytic Proof of Distance Estimation.................... 54 Chapter 5 Quaternions and the Dirac String Trick 57 Part 3 Hypercomplex Iterations 63 Chapter 6 Quaternion Mandelbrot Sets 65 6.1. Quaternion Mandelbrot Sets................ 65 6.2. The Distance Estimate for Quaternion Mandelbrot Sets....................... 65 Chapter 7 Distance Estimation in Higher Dimensional Spaces 71 7.1. Higher Dimensional Deterministic Fractals........ 71 7.2. The Cayley Numbers.................... 73 7.3. Distance Estimation in Higher Dimensional Spaces ... 74 7.4. Calculating the Derivative in Higher Dimensional Space............................. 77 7.5. Another Version of the Distance Estimation Formula . . 82
Contents IX Part 4 Inverse Iteration, Ray Tracing and Virtual Reality 89 Chapter 8 Inverse Iteration: An Interactive Visualization 91 8.1. Classical Inverse Iteration................. 91 8.2. Mappings in the Quaternions............... 93 8.3. The Quaternion Square Root ............... 94 8.4. The nth Roots in Higher Dimensions........... 96 8.5. Quaternion Julia Sets via Inverse Iteration........ 97 8.6. Functions Used in the Inverse Iteration Method..... 98 8.7. An Algorithm for the Inverse Iteration Method.....100 8.8. Tree Pruning........................102 8.9. Displaying Julia Sets....................104 Chapter 9 Ray Tracing Methods by Distance Estimation 107 9.1. Distance Estimation via Ray Tracing...........107 9.2. A Classical Ray Tracing Algorithm............108 9.3. A Ray Tracing Algorithm Using Distance Estimation..........................108 9.4. Quaternion Multiplication in the Algorithm.......110 9.5. Calculating the Derivative in the Algorithm.......Ill 9.6. Some Important Parameters in the Algorithm......113 9.7. The nth power Family of Quaternion Mandelbrot Sets .............................114 9.8. The Quadratic Family of Julia Sets............116 9.9. Generalized Quaternion Julia Sets.............118 9.10. Disconnected Quaternion Julia Sets............121 9.11. Displaying and Rendering.................122 9.11.1. Light models....................122 9.11.2. Surface normal...................123 9.11.3. Clarity.......................125 9.11.4. Other Rendering Considerations.........127
X Contents Chapter 10 Quaternion Deterministic Fractals in Virtual Reality 129 10.1. Introduction to Virtual Reality..............129 10.2. Parallel Computation....................130 10.3. Data Communication....................131 10.4. An Improved Display Algorithm..............132 10.5. Display of Quaternion Deterministic Fractals in VR . . . 133 10.6. Conclusion..........................134 Appendix A 135 Appendix B _ 137 Bibliography 139 Index ~ 143
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| dewey-full | 511.4 |
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| id | DE-604.BV017102162 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T19:13:45Z |
| institution | BVB |
| isbn | 9810232969 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-010313648 |
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| owner_facet | DE-29T |
| physical | XV, 144 S. graph. Darst. 2 CD-ROMs (12 cm) |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | World Scientific |
| record_format | marc |
| series | Series on knots and everything |
| series2 | Series on knots and everything |
| spelling | Dang, Yumei Verfasser aut Hypercomplex iterations distance estimation and higher dimensional fractals Yumei Dang ; Louis H. Kauffman ; Daniel Sandin Singapore [u.a.] World Scientific 2002 XV, 144 S. graph. Darst. 2 CD-ROMs (12 cm) txt rdacontent n rdamedia nc rdacarrier Series on knots and everything 17 Fractales Itération (Mathématiques) Mandelbrot, Ensembles de Quaterions Fractals Iterative methods (Mathematics) Mandelbrot sets Quaternions Iteration (DE-588)4123457-1 gnd rswk-swf Visualisierung (DE-588)4188417-6 gnd rswk-swf Fraktal (DE-588)4123220-3 gnd rswk-swf Hyperkomplexe Funktion (DE-588)4161069-6 gnd rswk-swf Hyperkomplexe Funktion (DE-588)4161069-6 s Iteration (DE-588)4123457-1 s Fraktal (DE-588)4123220-3 s DE-604 Visualisierung (DE-588)4188417-6 s Kauffman, Louis H. 1945- Verfasser (DE-588)134212614 aut Sandin, Daniel Verfasser aut Series on knots and everything 17 (DE-604)BV004608347 17 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010313648&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dang, Yumei Kauffman, Louis H. 1945- Sandin, Daniel Hypercomplex iterations distance estimation and higher dimensional fractals Series on knots and everything Fractales Itération (Mathématiques) Mandelbrot, Ensembles de Quaterions Fractals Iterative methods (Mathematics) Mandelbrot sets Quaternions Iteration (DE-588)4123457-1 gnd Visualisierung (DE-588)4188417-6 gnd Fraktal (DE-588)4123220-3 gnd Hyperkomplexe Funktion (DE-588)4161069-6 gnd |
| subject_GND | (DE-588)4123457-1 (DE-588)4188417-6 (DE-588)4123220-3 (DE-588)4161069-6 |
| title | Hypercomplex iterations distance estimation and higher dimensional fractals |
| title_auth | Hypercomplex iterations distance estimation and higher dimensional fractals |
| title_exact_search | Hypercomplex iterations distance estimation and higher dimensional fractals |
| title_full | Hypercomplex iterations distance estimation and higher dimensional fractals Yumei Dang ; Louis H. Kauffman ; Daniel Sandin |
| title_fullStr | Hypercomplex iterations distance estimation and higher dimensional fractals Yumei Dang ; Louis H. Kauffman ; Daniel Sandin |
| title_full_unstemmed | Hypercomplex iterations distance estimation and higher dimensional fractals Yumei Dang ; Louis H. Kauffman ; Daniel Sandin |
| title_short | Hypercomplex iterations |
| title_sort | hypercomplex iterations distance estimation and higher dimensional fractals |
| title_sub | distance estimation and higher dimensional fractals |
| topic | Fractales Itération (Mathématiques) Mandelbrot, Ensembles de Quaterions Fractals Iterative methods (Mathematics) Mandelbrot sets Quaternions Iteration (DE-588)4123457-1 gnd Visualisierung (DE-588)4188417-6 gnd Fraktal (DE-588)4123220-3 gnd Hyperkomplexe Funktion (DE-588)4161069-6 gnd |
| topic_facet | Fractales Itération (Mathématiques) Mandelbrot, Ensembles de Quaterions Fractals Iterative methods (Mathematics) Mandelbrot sets Quaternions Iteration Visualisierung Fraktal Hyperkomplexe Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010313648&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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