Advances in Discrete Differential Geometry

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This book explores the emerging field of discrete differential geometry, highlighting its connections to complex analysis, integrable systems, and applications in computer graphics. The authors examine discrete models in differential geometry and dynamical systems, where curves are polygonal, surfaces consist of triangles and quadrilaterals, and time is discrete. Despite this, the differences from smooth curves, surfaces, and classical dynamical systems are nearly imperceptible, embodying the concept of structure-preserving discretizations. Advances in this field are largely driven by its relevance to computer graphics and mathematical physics. Written by specialists collaborating on a research project, the book covers differential geometry, dynamical systems, and both smooth and discrete theories, alongside pure mathematics and its applications. Concrete examples illustrate the interaction of these areas, including discrete conformal mappings, discrete complex analysis, discrete curvatures, special surfaces, discrete integrable systems, and conformal texture mappings in computer graphics, as well as free-form architecture. Richly illustrated, the book demonstrates the beauty and utility of this new branch of mathematics, appealing to graduate students and researchers in various fields, including differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, and computer graphics.