Computational conformal geometry:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Somerville, Mass.
International Pr.
2008
Beijing Higher Education Press |
| Series: | Advanced lectures in mathematics
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IV, 295 S. Ill., graph. Darst. 1 CD |
| ISBN: | 9781571461711 |
Staff View
MARC
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| 245 | 1 | 0 | |a Computational conformal geometry |c Eds.: Xiangfeng David Gu & Shing-Tung Yau |
| 264 | 1 | |a Somerville, Mass. |b International Pr. |c 2008 | |
| 264 | 1 | |a Beijing |b Higher Education Press | |
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Record in the Search Index
| _version_ | 1804140646352551936 |
|---|---|
| adam_text | Titel: Computational conformal geometry
Autor: Gu, Xianfeng David
Jahr: 2008
Contents
1 Introduction........................................................ 1
1.1 Overview of Theories............................................ 3
1.1.1 Riemann Mapping........................................ 4
1.1.2 Riemann Uniformization................................... 5
1.1.3 Shape Space............................................. 6
1.1.4 General Geometric Structure................................ 7
1.2 Algorithms for Computing Conformai Mappings..................... 9
1.3 Applications.................................................... 14
1.3.1 Computer Graphics ....................................... 16
1.3.2 Computer Vision.......................................... 19
1.3.3 Geometric Modeling...................................... 25
1.3.4 Medical Imaging.......................................... 27
Further Readings..................................................... 29
Parti Theories
2 Homotopy Group................................................... 33
2.1 Algebraic Topological Methodology................................ 33
2.2 Surface Topological Classification................................. 35
2.3 Homotopy of Continuous Mappings................................ 40
2.4 Homotopy Group................................................ 41
2.5 Homotopy Invariant............................................. 42
2.6 Covering Spaces................................................ 43
2.7 Group Representation............................................ 46
2.8 Seifert-van Kampen Theorem..................................... 47
Problems........................................................... 49
3 Homology and Cohomology.......................................... 51
3.1 Simplicial Homology............................................ 51
3.1.1 Simplicial Complex....................................... 51
3.1.2 Geometric Approximation Accuracy......................... 52
II Contents
3.1.3 Chain Complex........................................... 55
3.1.4 Chain Map and Induced Homomorphism..................... 58
3.1.5 Simplicial Map........................................... 59
3.1.6 Chain Homotopy......................................... 60
.7 Homotopy Equivalence.................................... 60
.8 Relation Between Homology Group and Homotopy Group...... 61
.9 Lefschetz Fixed Point..................................... 62
.10 Mayer-Vietoris Homology Sequence ........................ 63
1.11 Tunnel Loop and Handle Loop.............................. 64
3.2 Cohomology................................................... 65
3.2.1 Cohomology Group....................................... 66
3.2.2 Cochain Map............................................. 67
3.2.3 Cochain Homotopy....................................... 68
Problems........................................................... 69
4 Exterior Differential Calculus........................................ 71
4.1 Smooth Manifold ............................................... 71
4.2 Differential Forms............................................... 73
4.3 Integration..................................................... 74
4.4 Exterior Derivative and Stokes Theorem............................ 75
4.5 De Rham Cohomology Group..................................... 75
4.6 Harmonic Forms................................................ 77
4.7 Hodge Theorem................................................. 78
Problems........................................................... 79
5 Differential Geometry of Surfaces..................................... 81
5.1 Curve Theory................................................... 81
5.2 Local Theory of Surfaces......................................... 84
5.2.1 Regular Surface .......................................... 84
5.2.2 First Fundamental Form ................................... 85
5.2.3 Second Fundamental Form................................. 86
5.2.4 Weingarten Transformation................................. 87
5.3 Orthonormal Movable Frame...................................... 88
5.3.1 Structure Equation........................................ 90
5.4 Covariant Differentiation......................................... 93
5.4.1 Geodesic Curvature....................................... 94
5.5 Gauss-Bonnet Theorem.......................................... 95
5.6 Index Theorem of Tangent Vector Field............................. 96
5.7 Minimal Surface................................................ 98
5.7.1 Weierstrass Representation................................. 100
5.7.2 Costa Minimal Surface.................................... 102
Problems........................................................... 102
Contents ??
Riemann Surface....................................................107
6.1 Riemann Surface................................................107
6.2 Riemann Mapping Theorem.......................................112
6.2.1 Conformai Module........................................113
6.2.2 Quasi-Conformal Mapping.................................114
6.2.3 Holomorphic Mappings....................................114
6.3 Holomorphic One-Forms.........................................116
6.4 Period Matrix...................................................118
6.5 Riemann-Roch Theorem..........................................121
6.6 Abel Theorem..................................................124
6.7 Uniformization .................................................125
6.8 Hyperbolic Riemann Surface......................................127
6.9 Teichmüller Space...............................................130
6.9.1 Quasi-Conformal Map.....................................130
6.9.2 Extremal Quasi-Conformal Map.............................132
6.10 Teichmüller Space and Modular Space .............................133
6.10.1 Fricke Space Model.......................................134
6.10.2 Geodesic Spectrum........................................136
Problems...........................................................137
Harmonic Maps and Surface Ricci Flow...............................139
7.1 Harmonic Maps of Surfaces.......................................139
7.1.1 Harmonic Energy and Harmonic Maps.......................140
7.1.2 Harmonic Map Equation...................................141
7.1.3 Radó s Theorem..........................................141
7.1.4 Hopf Differential.........................................142
7.1.5 Complex Form...........................................143
7.1.6 Bochner Formula.........................................143
7.1.7 Existence and Regularity...................................145
7.1.8 Uniqueness..............................................146
7.2 Surface Ricci How..............................................147
7.2.1 Conformai Deformation....................................147
7.2.2 Surface Ricci How........................................149
Problems...........................................................150
Geometric Structure.................................................153
8.1 (X,G) Geometric Structure......................................154
8.2 Development and Holonomy......................................154
8.3 Affine Structures on Surfaces......................................155
8.4 Spherical Structure.............................................. 56
8.5 Euclidean Structure..............................................157
8.6 Hyperbolic Structure.............................................159
8.7 Real Projective Structure.........................................160
Problems...........................................................161
IV Contents
Part II Algorithms
9 Topological Algorithms..............................................167
9.1 Triangular Meshes...............................................167
9.1.1 Half-Edge Data Structure..................................168
9.1.2 Code Samples............................................171
9.2 Cut Graph......................................................176
9.3 Fundamental Domain............................................178
9.4 Basis of Homotopy Group........................................179
9.5 Gluing Two Meshes .............................................180
9.6 Universal Covering Space ........................................181
9.7 Curve Lifting...................................................183
9.8 Homotopy Detection.............................................185
9.9 The Shortest Loop...............................................186
9.10 Canonical Homotopy Group Generator.............................187
Further Readings.....................................................190
Problems...........................................................190
10 Algorithms for Harmonic Maps.......................................193
10.1 Piecewise Linear Functional Space, Inner Product and Laplacian........194
10.2 Newton s Method for Open Surface................................198
10.3 Non-Linear Heat Diffusion for Closed Surfaces......................200
10.4 Riemann Mapping...............................................204
10.5 Least Square Method for Solving Beltrami Equation..................206
10.6 General Surface Mapping.........................................208
Further Readings.....................................................212
Problems...........................................................212
11 Harmonic Forms and Holomorphic Forms.............................215
11.1 Characteristic Forms.............................................216
11.2 Wedge Product..................................................217
11.3 Characteristic 1-Form............................................218
11.4 Computing Cohomology Basis....................................219
11.5 Harmonic 1-Form...............................................221
11.6 Hodge Star Operator.............................................222
11.7 Holomorphic 1-Form............................................224
11.8 Inner Product Among 1-Forms....................................228
11.9 Holomorphic Forms on Surfaces with Boundaries....................229
11.10 Zero Points and Critical Trajectories...............................232
11.11 Flat Metric Induced by Holomorphic 1-Forms.......................235
11.12 Conformai Invariants............................................238
11.13 Conformai Mappings for Multi-Holed Annuli.......................241
Further Readings.....................................................243
Problems...........................................................244
Contents V
12 Discrete Ricci Flow..................................................247
12.1 Circle Packing Metric............................................248
12.2 Discrete Gaussian Curvature......................................251
12.3 Discrete Surface Ricci Flow.......................................254
12.4 Newton s Method...............................................257
12.5 Isometric Planar Embedding......................................260
12.6 Surfaces with Boundaries.........................................262
12.7 Optimal Parameterization Using Ricci Flow.........................264
12.8 Hyperbolic Ricci Flow...........................................267
12.9 Hyperbolic Embedding...........................................269
12.9.1 Poincaré Disk Model......................................269
12.9.2 Embedding the Fundamental Domain........................270
12.9.3 Hyperbolic Embedding of the Universal Covering Space........271
12.10 Hyperbolic Ricci Flow for Surfaces with Boundaries.................274
Further Readings.....................................................276
Problems...........................................................276
A Major Algorithms...................................................281
? Acknowledgement...................................................283
Reference ..............................................................285
Index..................................................................291
|
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| author | Gu, Xianfeng David |
| author_GND | (DE-588)174021356 |
| author_facet | Gu, Xianfeng David |
| author_role | aut |
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| ctrlnum | (OCoLC)276910894 (DE-599)OBVAC07055153 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.223 |
| dewey-search | 515.223 |
| dewey-sort | 3515.223 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T22:03:17Z |
| institution | BVB |
| isbn | 9781571461711 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018596397 |
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| owner | DE-83 DE-91G DE-BY-TUM |
| owner_facet | DE-83 DE-91G DE-BY-TUM |
| physical | IV, 295 S. Ill., graph. Darst. 1 CD |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | International Pr. Higher Education Press |
| record_format | marc |
| series | Advanced lectures in mathematics |
| series2 | Advanced lectures in mathematics |
| spelling | Gu, Xianfeng David Verfasser (DE-588)174021356 aut Computational conformal geometry Eds.: Xiangfeng David Gu & Shing-Tung Yau Somerville, Mass. International Pr. 2008 Beijing Higher Education Press IV, 295 S. Ill., graph. Darst. 1 CD txt rdacontent n rdamedia nc rdacarrier Advanced lectures in mathematics 3 Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Konforme Differentialgeometrie (DE-588)4206468-5 gnd rswk-swf Konforme Differentialgeometrie (DE-588)4206468-5 s Algorithmische Geometrie (DE-588)4130267-9 s DE-604 Advanced lectures in mathematics 3 (DE-604)BV024628521 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018596397&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gu, Xianfeng David Computational conformal geometry Advanced lectures in mathematics Algorithmische Geometrie (DE-588)4130267-9 gnd Konforme Differentialgeometrie (DE-588)4206468-5 gnd |
| subject_GND | (DE-588)4130267-9 (DE-588)4206468-5 |
| title | Computational conformal geometry |
| title_auth | Computational conformal geometry |
| title_exact_search | Computational conformal geometry |
| title_full | Computational conformal geometry Eds.: Xiangfeng David Gu & Shing-Tung Yau |
| title_fullStr | Computational conformal geometry Eds.: Xiangfeng David Gu & Shing-Tung Yau |
| title_full_unstemmed | Computational conformal geometry Eds.: Xiangfeng David Gu & Shing-Tung Yau |
| title_short | Computational conformal geometry |
| title_sort | computational conformal geometry |
| topic | Algorithmische Geometrie (DE-588)4130267-9 gnd Konforme Differentialgeometrie (DE-588)4206468-5 gnd |
| topic_facet | Algorithmische Geometrie Konforme Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018596397&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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