Advances in Discrete Differential Geometry

1. Front Matter

   Pages i-x

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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. Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

   • Ulrike Bücking

   Pages 133-149Open Access

5. Numerical Methods for the Discrete Map \(Z^a\)

   • Folkmar Bornemann, Alexander Its, Sheehan Olver, Georg Wechslberger

   Pages 151-176Open Access

6. A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths

   • Hana Kouřimská, Lara Skuppin, Boris Springborn

   Pages 177-195Open Access

7. Complex Line Bundles Over Simplicial Complexes and Their Applications

   • Felix Knöppel, Ulrich Pinkall

   Pages 197-239Open Access

8. Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

   • Wai Yeung Lam, Ulrich Pinkall

   Pages 241-265Open Access

9. Vertex Normals and Face Curvatures of Triangle Meshes

   • Xiang Sun, Caigui Jiang, Johannes Wallner, Helmut Pottmann

   Pages 267-286Open Access

10. S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature

    • Alexander I. Bobenko, Tim Hoffmann

    Pages 287-308Open Access

11. Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization

    • Ulrike Bücking, Daniel Matthes

    Pages 309-345Open Access

12. On the Lagrangian Structure of Integrable Hierarchies

    • Yuri B. Suris, Mats Vermeeren

    Pages 347-378Open Access

13. On the Variational Interpretation of the Discrete KP Equation

    • Raphael Boll, Matteo Petrera, Yuri B. Suris

    Pages 379-405Open Access

14. Six Topics on Inscribable Polytopes

    • Arnau Padrol, Günter M. Ziegler

    Pages 407-419Open Access

15. 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

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

• Technical University of Berlin, Berlin, Germany

  Alexander I. Bobenko

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“.