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  • Book
  • Open Access
  • © 2016

Advances in Discrete Differential Geometry

  • One of the first books on a newly emerging field of discrete differential geometry

  • Excellent way to access this new exciting area

  • Topics survey fascinating connections of discrete models in differential geometry to complex analysis, integrable systems, and to applications in computer graphics

  • Includes supplementary material:

Buying options

Softcover Book USD 54.99
Price excludes VAT (USA)
Hardcover Book USD 59.99
Price excludes VAT (USA)

Table of contents (14 chapters)

  1. Front Matter

    Pages i-x
  2. Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization

    • Alexander I. Bobenko, Stefan Sechelmann, Boris Springborn
    Pages 1-56Open Access
  3. Discrete Complex Analysis on Planar Quad-Graphs

    • Alexander I. Bobenko, Felix Günther
    Pages 57-132Open Access
  4. Numerical Methods for the Discrete Map \(Z^a\)

    • Folkmar Bornemann, Alexander Its, Sheehan Olver, Georg Wechslberger
    Pages 151-176Open Access
  5. A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths

    • Hana Kouřimská, Lara Skuppin, Boris Springborn
    Pages 177-195Open Access
  6. Complex Line Bundles Over Simplicial Complexes and Their Applications

    • Felix Knöppel, Ulrich Pinkall
    Pages 197-239Open Access
  7. Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

    • Wai Yeung Lam, Ulrich Pinkall
    Pages 241-265Open Access
  8. Vertex Normals and Face Curvatures of Triangle Meshes

    • Xiang Sun, Caigui Jiang, Johannes Wallner, Helmut Pottmann
    Pages 267-286Open Access
  9. On the Lagrangian Structure of Integrable Hierarchies

    • Yuri B. Suris, Mats Vermeeren
    Pages 347-378Open Access
  10. On the Variational Interpretation of the Discrete KP Equation

    • Raphael Boll, Matteo Petrera, Yuri B. Suris
    Pages 379-405Open Access
  11. Six Topics on Inscribable Polytopes

    • Arnau Padrol, Günter M. Ziegler
    Pages 407-419Open Access
  12. DGD Gallery: Storage, Sharing, and Publication of Digital Research Data

    • Michael Joswig, Milan Mehner, Stefan Sechelmann, Jan Techter, Alexander I. Bobenko
    Pages 421-439Open Access

About this book

This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics.

The authors take a closer look at discrete models in differential

geometry and dynamical systems. Their curves are polygonal, surfaces

are made from triangles and quadrilaterals, and time is discrete.

Nevertheless, the difference between the corresponding smooth curves,

surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, and on pure mathematics and its practical applications. The interaction of these facets is demonstrated by concrete examples, including discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, conformal texture mappings in computer graphics, and free-form architecture.

This richly illustrated book will convince readers that this new branch of mathematics is both beautiful and useful. It will appeal to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, as well as computer graphics and geometry processing.


  • discrete conformal maps
  • discrete curvature
  • integrable systems
  • discrete complex analysis
  • triangle meshes
  • polyhedral surfaces

Editors and Affiliations

  • Technical University of Berlin, Berlin, Germany

    Alexander I. Bobenko

About the editor

Alexander I. Bobenko is a professor at the Technische Universität Berlin. He is an author with Yuri Suris of the book „Discrete Differential Geometry“, editor of several books on this topic and a co-organizer of the regular conference „Discrete Differential Geometry“ in Oberwolfach. He is the Coordinator of the DFG Collaboration Research Center „Discretization in Geometry and Dynamics“.

Bibliographic Information

Buying options

Softcover Book USD 54.99
Price excludes VAT (USA)
Hardcover Book USD 59.99
Price excludes VAT (USA)