Discrete differential geometry: integrable structure
An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from...
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Providence, RI
American Mathematical Society
2008
|
| Schriftenreihe: | Graduate studies in mathematics
98 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best?This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results.The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question 'How do we discretize differential geometry?' arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful. |
| Beschreibung: | XXIV, 404 S. graph. Darst. |
| ISBN: | 9780821847008 0821847007 |
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| 100 | 1 | |a Bobenko, Aleksandr I. |d 1960- |e Verfasser |0 (DE-588)122463625 |4 aut | |
| 245 | 1 | 0 | |a Discrete differential geometry |b integrable structure |c Alexander I. Bobenko ; Yuri B. Suris |
| 264 | 1 | |a Providence, RI |b American Mathematical Society |c 2008 | |
| 300 | |a XXIV, 404 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 98 | |
| 520 | |a An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best?This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results.The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question 'How do we discretize differential geometry?' arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful. | ||
| 650 | 4 | |a Discrete geometry | |
| 650 | 4 | |a Geometry, Differential | |
| 650 | 4 | |a Integral geometry | |
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Datensatz im Suchindex
| _version_ | 1804138603566071808 |
|---|---|
| adam_text | Contents
Preface
■ .;. ■ xi
Introduction
xiii
What is discrete differential geometry
xiii
Integrability
> xv
Prom discrete to smooth
xvii
Structure of this book
xxi
How to read this book
xxii
Acknowledgements
, . xxiii
Chapter
1.
Classical Differential Geometry
■...:■■ ■■[ 1
1.1.
Conjugate nets
>.-■ ■ 2
1.1.1.
Notion of conjugate nets
■;■/„, * 2
1.1.2.
Alternative analytic description of conjugate nets
:. , 3
1.1.3.
Transformations of conjugate nets
, <· 4
1.1.4.
Classical formulation of F-transformation
5
1.2.
Koenigs and
Moutard
nets
7
1.2.1.
Notion of Koenigs and
Moutard
nets
7
1.2.2.
Transformations of Koenigs and
Moutard
nets
9
1.2.3.
Classical formulation of the
Moutard
transformation
10
1.3.
Asymptotic nets
·
řií
ч:
11
1.4.
Orthogonal nets
12
1.4.1.
Notion of orthogonal nets
. . .·. ■:■··;■}* 12
1.4.2.
Analytic description of orthogonal nets
14
1.4.3.
Spinor frames of orthogonal nets
15
1.4.4.
Curvatures of surfaces and curvature line
parametrized surfaces
í
iß
vi
Contents
1.4.5.
Ribaucour
transformations
of orthogonal nets
17
1.5.
Principally parametrized sphere congruences
19
1.6.
Surfaces with constant negative Gaussian curvature
20
1.7.
Isothermic surfaces
22
1.8.
Surfaces with constant mean curvature
26
1.9.
Bibliographical notes
28
Chapter
2.
Discretization Principles. Multidimensional Nets
31
2.1.
Discrete conjugate nets (Q-nets)
32
2.1.1.
Notion and consistency of Q-nets
32
2.1.2.
Transformations of Q-nets
38
2.1.3.
Alternative analytic description of Q-nets
40
2.1.4.
Continuous limit
42
2.2.
Discrete line congruences
43
2.3.
Discrete Koenigs and
Moutard
nets
47
2.3.1.
Notion of dual quadrilaterals
47
2.3.2.
Notion of discrete Koenigs nets
49
2.3.3.
Geometric characterization of two-dimensional
discrete Koenigs nets
54
2.3.4.
Geometric characterization of three-dimensional
discrete Koenigs nets
56
2.3.5.
Function
v
and
Christoffel
duality
58
2.3.6.
Moutard
representative of a discrete Koenigs net
60
2.3.7.
Continuous limit
60
2.3.8.
Notion and consistency of T-nets
61
2.3.9.
Transformations of T-nets
63
2.3.10.
Discrete M-nets
65
2.4.
Discrete asymptotic nets
66
2
A.I. Notion and consistency of discrete asymptotic nets
66
2.4.2.
Discrete Lelieuvre representation
70
2.4.3.
Transformations of discrete A-nets
72
2.5.
Exercises
73
2.6.
Bibliographical notes
82
Chapter
3.
Discretization Principles. Nets in Quadrics
87
3.1.
Circular nets
88
3.1.1.
Notion and consistency of circular nets
88
3.1.2.
Transformations of circular nets
92
3.1.3.
Analytic description of circular nets
93
3.1.4.
Möbius-geometric
description of circular nets
96
Contents
г
vii
3.2.
Q-nets in quadrics
99
3.3.
Discrete line
congruences in
quadrics
101
3.4.
Conical nets
103
3.5.
Principal
contact element nets
106
3.6.
Q-congruences of spheres
110
3.7.
Ribaucour congruences of spheres
113
3.8.
Discrete curvature line parametrization in Lie,
Möbius
and
Laguerre geometries
115
3.9.
Discrete asymptotic nets in
Plücker
line geometry
118
3.10.
Exercises
120
3.11.
Bibliographical notes
123
Chapter
4.
Special Classes of Discrete Surfaces
127
4.1.
Discrete
Moutard
nets in quadrics
127
4.2.
Discrete K-nets
130
4.2.1.
Notion of a discrete K-net
130
4.2.2.
Bäcklund
transformation
133
4.2.3.
Hirota equation
133
4.2.4.
Discrete zero curvature representation
139
4.2.5.
Discrete K-surfaces
139
4.2.6.
Discrete sine-Gordon equation
142
4.3.
Discrete isothermic nets
145
4.3.1.
Notion of a discrete isothermic net
145
4.3.2.
Cross-ratio characterization of discrete isothermic
nets
147
4.3.3.
Darboux transformation of discrete isothermic nets
151
4.3.4.
Metric of a discrete isothermic net
152
4.3.5.
Moutard
representatives of discrete isothermic nets
155
4.3.6. Christoffel
duality for discrete isothermic nets
156
4.3.7. 3D
consistency and zero curvature representation
158
4.3.8.
Continuous limit
160
4.4.
S-isothermic nets
161
4.5.
Discrete surfaces with constant curvature
170
4.5.1.
Parallel discrete surfaces and line congruences
170
4.5.2.
Polygons with parallel edges and mixed area
170
4.5.3.
Curvatures of a polyhedral surface with a parallel
Gauss map
173
4.5.4.
Q-nets with constant curvature
175
4.5.5.
Curvature of principal contact element nets
177
viii Contents
4.5.6.
Circular minimal
nets and nets with constant mean
curvature
178
4.6.
Exercises
179
4.7.
Bibliographical notes
183
Chapter
5.
Approximation
187
5.1.
Discrete hyperbolic systems
187
5.2.
Approximation in discrete hyperbolic systems
190
5.3.
Convergence of Q-nets
196
5.4.
Convergence of discrete
Moutard
nets
197
5.5.
Convergence of discrete asymptotic nets
199
5.6.
Convergence of circular nets
200
5.7.
Convergence of discrete K-surfaces
205
5.8.
Exercises
206
5.9.
Bibliographical notes
207
Chapter
6.
Consistency as Integrability
209
6.1.
Continuous
integrable
systems
210
6.2.
Discrete
integrable
systems
213
6.3.
Discrete 2D
integrable
systems on graphs
215
6.4.
Discrete Laplace type equations
217
6.5.
Quad-graphs
218
6.6.
Three-dimensional consistency
220
6.7.
From
3D
consistency to zero curvature representations and
Bäcklund
transformations
222
6.8.
Geometry of boundary value problems for
integrable 2D
equations
227
6.8.1.
Initial value problem
228
6.8.2.
Extension to a multidimensional lattice
231
6.9. 3D
consistent equations with
noncommutative
fields
235
6.10.
Classification of discrete
integrable 2D
systems with fields
on vertices. I
239
6.11.
Proof of the classification theorem
242
6.11.1. 3D
consistent systems, biquadratics and
tetrahedron property
242
6.11.2.
Analysis: descending from
multiaffine Q
to
quartic
r
245
6.11.3.
Synthesis: ascending from quartic
r
to biquadratic
h
247
Contents ix:
6.11.4.
Synthesis: ascending from biquadratics
/ι1·7
to
■ , ■{■■.:■
multiaffine Q 249
6.11.5.
Putting equations
Q
= 0
on the cube
251
6.12.
Classification of discrete
integrable 2D
systems with fields
*
on vertices. II
252
6.13.
Integrable
discrete Laplace type equations
256
6.14.
Fields on edges: Yang-Baxter maps
261
6.15.
Classification of Yang-Baxter maps
266
6.16.
Discrete
integrable
3D
systems
272
6.16.1.
Fields on 2-faces.
272
6.16.2.
Fields on vertices.
276
6.17.
Exercises
279
6.18.
Bibliographical notes
286
Chapter
7.
Discrete Complex Analysis. Linear Theory
291
7.1.
Basic notions of discrete linear complex analysis
291
7.2.
Moutard
transformation for discrete Cauchy-Riemann
equations
294
7.3.
Integrable
discrete Cauchy-Riemann equations
297
7.4.
Discrete exponential functions
300
7.5.
Discrete logarithmic function !
302
7.6.
Exercises
307
7.7.
Bibliographical notes
308
Chapter
8.
Discrete Complex Analysis.
Integrable
Circle Patterns
311
8.1.
Circle patterns
311
8.2.
Integrable
cross-ratio and Hirota systems
313
8.3.
Integrable
circle patterns
316
8.4.
za and log
z
circle patterns
319
8.5.
Linearization
324
8.6.
Exercises
326
8.7.
Bibliographical notes
327
Chapter
9.
Foundations
331
9.1.
Projective
geometry
331
9.2.
Lie geometry
335
9.2.1.
Objects of Lie geometry
335
9.2.2.
Projective
model of Lie geometry
336
9.2.3.
Lie sphere transformations
339
χ
Contents
9.2.4.
Planar
families
of spheres;
Dupin
cyclides
340
9.3.
Möbius
geometry
341
9.3.1.
Objects of
Möbius
geometry
341
9.3.2.
Projective
model of
Möbius
geometry
344
9.3.3.
Möbius
transformations
348
9.4.
Laguerre geometry
350
9.5. Plücker
line geometry
353
9.6.
Incidence theorems
357
9.6.1.
Menelaus
and Ceva s theorems
357
9.6.2.
Generalized
Menelaus
theorem
360
9.6.3.
Desargues
theorem
361
9.6.4.
Quadrangular sets
362
9.6.5.
Carnot s and Pascal s theorems
364
9.6.6.
Brianchon s theorem
366
9.6.7.
Miquel s theorem
367
Appendix. Solutions of Selected Exercises
369
A.I. Solutions of exercises to Chapter
2 369
A.2. Solutions of exercises to Chapter
3 376
A.3. Solutions of exercises to Chapter
4 377
A.4. Solutions of exercises to Chapter
6 381
Bibliography
385
Notation
399
Index
401
|
| any_adam_object | 1 |
| author | Bobenko, Aleksandr I. 1960- Suris, Juri B. 1963- |
| author_GND | (DE-588)122463625 (DE-588)1051230527 |
| author_facet | Bobenko, Aleksandr I. 1960- Suris, Juri B. 1963- |
| author_role | aut aut |
| author_sort | Bobenko, Aleksandr I. 1960- |
| author_variant | a i b ai aib j b s jb jbs |
| building | Verbundindex |
| bvnumber | BV035302556 |
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| classification_tum | MAT 530f MAT 052f |
| ctrlnum | (OCoLC)475243803 (DE-599)HBZHT015821598 |
| dewey-full | 516.3 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 516 - Geometry |
| dewey-raw | 516.3 |
| dewey-search | 516.3 |
| dewey-sort | 3516.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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Bobenko ; Yuri B. Suris</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="a">Providence, RI</subfield><subfield code="b">American Mathematical Society</subfield><subfield code="c">2008</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">XXIV, 404 S.</subfield><subfield code="b">graph. Darst.</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="1" ind2=" "><subfield code="a">Graduate studies in mathematics</subfield><subfield code="v">98</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best?This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results.The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question 'How do we discretize differential geometry?' arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). 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| id | DE-604.BV035302556 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:30:49Z |
| institution | BVB |
| isbn | 9780821847008 0821847007 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017107396 |
| oclc_num | 475243803 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-20 DE-83 DE-188 DE-355 DE-BY-UBR DE-739 DE-11 |
| owner_facet | DE-91G DE-BY-TUM DE-20 DE-83 DE-188 DE-355 DE-BY-UBR DE-739 DE-11 |
| physical | XXIV, 404 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Bobenko, Aleksandr I. 1960- Verfasser (DE-588)122463625 aut Discrete differential geometry integrable structure Alexander I. Bobenko ; Yuri B. Suris Providence, RI American Mathematical Society 2008 XXIV, 404 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 98 An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. For a given smooth geometry one can suggest many different discretizations. Which one is the best?This book answers this question by providing fundamental discretization principles and applying them to numerous concrete problems. It turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. The intended audience of this book is threefold. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. On the other hand, it is addressed to specialists in geometry and mathematical physics. It reflects the recent progress in discrete differential geometry and contains many original results.The third group of readers at which this book is targeted is formed by specialists in geometry processing, computer graphics, architectural design, numerical simulations, and animation. They may find here answers to the question 'How do we discretize differential geometry?' arising in their specific field. Prerequisites for reading this book include standard undergraduate background (calculus and linear algebra). No knowledge of differential geometry is expected, although some familiarity with curves and surfaces can be helpful. Discrete geometry Geometry, Differential Integral geometry Diskrete Geometrie (DE-588)4130271-0 gnd rswk-swf Differentialgeometrie (DE-588)4012248-7 gnd rswk-swf Diskrete Geometrie (DE-588)4130271-0 s Differentialgeometrie (DE-588)4012248-7 s DE-604 Suris, Juri B. 1963- Verfasser (DE-588)1051230527 aut Erscheint auch als Online-Ausgabe 978-1-4704-1793-2 Graduate studies in mathematics 98 (DE-604)BV009739289 98 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017107396&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bobenko, Aleksandr I. 1960- Suris, Juri B. 1963- Discrete differential geometry integrable structure Graduate studies in mathematics Discrete geometry Geometry, Differential Integral geometry Diskrete Geometrie (DE-588)4130271-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| subject_GND | (DE-588)4130271-0 (DE-588)4012248-7 |
| title | Discrete differential geometry integrable structure |
| title_auth | Discrete differential geometry integrable structure |
| title_exact_search | Discrete differential geometry integrable structure |
| title_full | Discrete differential geometry integrable structure Alexander I. Bobenko ; Yuri B. Suris |
| title_fullStr | Discrete differential geometry integrable structure Alexander I. Bobenko ; Yuri B. Suris |
| title_full_unstemmed | Discrete differential geometry integrable structure Alexander I. Bobenko ; Yuri B. Suris |
| title_short | Discrete differential geometry |
| title_sort | discrete differential geometry integrable structure |
| title_sub | integrable structure |
| topic | Discrete geometry Geometry, Differential Integral geometry Diskrete Geometrie (DE-588)4130271-0 gnd Differentialgeometrie (DE-588)4012248-7 gnd |
| topic_facet | Discrete geometry Geometry, Differential Integral geometry Diskrete Geometrie Differentialgeometrie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017107396&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT bobenkoaleksandri discretedifferentialgeometryintegrablestructure AT surisjurib discretedifferentialgeometryintegrablestructure |