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Discrete Differential Geometry

  • Book
  • © 2008

Overview

Editors:
  1. Alexander I. Bobenko

    1. Institut für Mathematik, MA 8-3, Technische Universität Berlin, Berlin, Germany

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  2. John M. Sullivan

    1. Institut für Mathematik, MA 3-2, Technische Universität Berlin, Berlin, Germany

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  3. Peter Schröder

    1. Department of Computer Science, Caltech, MS 256-80, Pasadena, USA

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  4. Günter M. Ziegler

    1. Institut für Mathematik, MA 6-2, Technische Universität Berlin, Berlin, Germany

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  • First book on a newly emerging field of discrete differential geometry, provides an excellent way to access this new exciting area
  • Carefully edited collection of essays by key researchers in the field
  • Surveys fascinating connections between discrete geometry and differential geometry: for instance circle patterns and triangulated surfaces on the discrete side connect to minimal surfaces and curvature line parametrizations on the differential side
  • Includes supplementary material: sn.pub/extras

Part of the book series: Oberwolfach Seminars (OWS, volume 38)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-x
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  2. Discretization of Surfaces: Special Classes and Parametrizations

    1. Front Matter

      Pages 1-1
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    2. Surfaces from Circles

      • Alexander I. Bobenko
      Pages 3-35

    3. Minimal Surfaces from Circle Patterns: Boundary Value Problems, Examples

      • Ulrike Bücking
      Pages 37-56

    4. Designing Cylinders with Constant Negative Curvature

      • Ulrich Pinkall
      Pages 57-66

    5. On the Integrability of Infinitesimal and Finite Deformations of Polyhedral Surfaces

      • Wolfgang K. Schief, Alexander I. Bobenko, Tim Hoffmann
      Pages 67-93

    6. Discrete Hashimoto Surfaces and a Doubly Discrete Smoke-Ring Flow

      • Tim Hoffmann
      Pages 95-115

    7. The Discrete Green’s Function

      • Yuri B. Suris
      Pages 117-133

  3. Curvatures of Discrete Curves and Surfaces

    1. Front Matter

      Pages 135-135
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    2. Curves of Finite Total Curvature

      • John M. Sullivan
      Pages 137-161

    3. Convergence and Isotopy Type for Graphs of Finite Total Curvature

      • Elizabeth Denne, John M. Sullivan
      Pages 163-174

    4. Curvatures of Smooth and Discrete Surfaces

      • John M. Sullivan
      Pages 175-188

  4. Geometric Realizations of Combinatorial Surfaces

    1. Front Matter

      Pages 189-189
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    2. Polyhedral Surfaces of High Genus

      • Günter M. Ziegler
      Pages 191-213

    3. Necessary Conditions for Geometric Realizability of Simplicial Complexes

      • Dagmar Timmreck
      Pages 215-233

    4. Enumeration and Random Realization of Triangulated Surfaces

      • Frank H. Lutz
      Pages 235-253

    5. On Heuristic Methods for Finding Realizations of Surfaces

      • Jürgen Bokowski
      Pages 255-260

  5. Geometry Processing and Modeling with Discrete Differential Geometry

    1. Front Matter

      Pages 261-261
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    2. What Can We Measure?

      • Peter Schröder
      Pages 263-273

    3. Convergence of the Cotangent Formula: An Overview

      • Max Wardetzky
      Pages 275-286
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Keywords

About this book

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.

Editors and Affiliations

  • Institut für Mathematik, MA 8-3, Technische Universität Berlin, Berlin, Germany

    Alexander I. Bobenko

  • Institut für Mathematik, MA 3-2, Technische Universität Berlin, Berlin, Germany

    John M. Sullivan

  • Department of Computer Science, Caltech, MS 256-80, Pasadena, USA

    Peter Schröder

  • Institut für Mathematik, MA 6-2, Technische Universität Berlin, Berlin, Germany

    Günter M. Ziegler

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