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Discrete Differential Geometry

✍ Scribed by Alexander I. Bobenko, Peter Schröder, John M. Sullivan, Günter M. Ziegler

Publisher
Birkhäuser Basel
Year
2008
Tongue
English
Leaves
337
Series
Oberwolfach Seminars
Edition
1
Category
Library
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✦ Synopsis

Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.

✦ Table of Contents

Mathematics - Discrete Differential Geometry - (Alexander I. Bobenko, Peter Schroder) Birkhauser 2008.pdf......Page 0
front-matter......Page 2
Preface......Page 6
Content......Page 9
Part I Discretization of Surfaces: Special Classes and Parametrizations......Page 11
Surfaces from Circles......Page 12
Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples......Page 45
Designing Cylinders with Constant Negative Curvature......Page 65
On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces......Page 75
Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow......Page 102
The Discrete Green’s Function......Page 123
Part II Curvatures of Discrete Curves and Surfaces......Page 140
Curves of Finite Total Curvature......Page 141
Convergence and Isotopy Type for Graphs of Finite Total Curvature......Page 166
Curvatures of Smooth and Discrete Surfaces......Page 178
Part III Geometric Realizations of Combinatorial Surfaces......Page 192
Polyhedral Surfaces of High Genus......Page 193
Necessary Conditions for Geometric Realizability of Simplicial Complexes......Page 216
Enumeration and Random Realization of Triangulated Surfaces......Page 235
On Heuristic Methods for Finding Realizations of Surfaces......Page 254
Part IV Geometry Processing and Modeling with Discrete Differential Geometry......Page 260
What Can We Measure?......Page 261
Convergence of the Cotangent Formula: An Overview......Page 272
Discrete Differential Forms for Computational Modeling......Page 284
A Discrete Model of Thin Shells......Page 322
back-matter......Page 335

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