Advances in Discrete Differential Geometry

by Alexander I. Bobenko

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Book Description

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.

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).

Table of Contents

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 Za
 
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

Book Details

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
CC BY-NC

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