Advances in discrete differential geometry

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This book introduces the emerging field of discrete differential geometry, exploring its connections to complex analysis, integrable systems, and applications in computer graphics. It focuses on discrete models in differential geometry and dynamical systems, featuring polygonal curves, triangular and quadrilateral surfaces, and discrete time. Despite their differences, these discrete models closely resemble their smooth counterparts in classical dynamical systems. The authors emphasize structure-preserving discretizations, highlighting the relevance of this field to computer graphics and mathematical physics. Written by specialists collaborating on a research project, the text delves into both smooth and discrete theories, showcasing pure mathematics alongside practical applications. Concrete examples illustrate the interplay between 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, this work demonstrates the beauty and utility of this new mathematical branch, appealing to graduate students and researchers in differential geometry, complex analysis, mathematical physics, numerical methods, discrete geometry, computer graphics, and geometry processing.