Discrete Differential Geometry

Alexander I. Bobenko (auth.), Alexander I. Bobenko, John M. Sullivan, Peter Schröder, Günter M. Ziegler (eds.)3764386207, 978-3-7643-8620-7

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 :
Front Matter….Pages i-x
Front Matter….Pages 1-1
Surfaces from Circles….Pages 3-35
Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples….Pages 37-56
Designing Cylinders with Constant Negative Curvature….Pages 57-66
On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces….Pages 67-93
Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow….Pages 95-115
The Discrete Green’s Function….Pages 117-133
Front Matter….Pages 135-135
Curves of Finite Total Curvature….Pages 137-161
Convergence and Isotopy Type for Graphs of Finite Total Curvature….Pages 163-174
Curvatures of Smooth and Discrete Surfaces….Pages 175-188
Front Matter….Pages 189-189
Polyhedral Surfaces of High Genus….Pages 191-213
Necessary Conditions for Geometric Realizability of Simplicial Complexes….Pages 215-233
Enumeration and Random Realization of Triangulated Surfaces….Pages 235-253
On Heuristic Methods for Finding Realizations of Surfaces….Pages 255-260
Front Matter….Pages 261-261
What Can We Measure?….Pages 263-273
Convergence of the Cotangent Formula: An Overview….Pages 275-286
Discrete Differential Forms for Computational Modeling….Pages 287-324
A Discrete Model of Thin Shells….Pages 325-337
Back Matter….Pages 339-341