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.

• Cover
• Frontmatter
• Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization
• Discrete Complex Analysis on Planar Quad-Graphs
• Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices
• Numerical Methods for the Discrete Map $$Z^a$$ Z a
• A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths
• Complex Line Bundles Over Simplicial Complexes and Their Applications
• Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes
• Vertex Normals and Face Curvatures of Triangle Meshes
• S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature
• Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization
• On the Lagrangian Structure of Integrable Hierarchies
• On the Variational Interpretation of the Discrete KP Equation
• Six Topics on Inscribable Polytopes
• DGD Gallery: Storage, Sharing, and Publication of Digital Research Data