**by Alexander I. Bobenko**

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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. ### Table of Contents

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This open book is licensed under a Creative Commons License (CC BY-NC). You can download Advances in Discrete Differential Geometry ebook for free in PDF format (29.0 MB).

Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization

1

Discrete Complex Analysis on Planar Quad-Graphs

57

Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices

133

Numerical Methods for the Discrete Map Z^{a}

151

A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths

177

Complex Line Bundles Over Simplicial Complexes and Their Applications

197

Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes

241

Vertex Normals and Face Curvatures of Triangle Meshes

267

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

287

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

309

On the Lagrangian Structure of Integrable Hierarchies

347

On the Variational Interpretation of the Discrete KP Equation

379

Six Topics on Inscribable Polytopes

407

DGD Gallery: Storage, Sharing, and Publication of Digital Research Data

421

Publisher

Springer

Published

2016

Pages

439

Edition

1

Language

English

ISBN13

9783662504468

ISBN10

3662504464

ISBN13 Digital

9783662504475

ISBN10 Digital

3662504472

PDF Size

29.0 MB

License

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