Aleksandr I. Bobenko Knihy

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.

Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Current progress in this field is to a large extent stimulated by its relevance for computer graphics and mathematical physics. This collection of essays, which documents the main lectures of the 2004 Oberwolfach Seminar on the topic, as well as a number of additional contributions by key participants, gives a lively, multi-facetted introduction to this emerging field.

This volume provides a structured overview of computational approaches to Riemann surfaces, showcasing both existing methods and those in development. Contributions come from groups that offer publicly available numerical codes, illustrating the software tools accessible for practical use. It includes examples of solutions to partial differential equations and surface theory. The intended audience is twofold: it serves as a textbook for graduate courses in the numerics of Riemann surfaces, requiring only a standard undergraduate background in calculus and linear algebra, with no prior knowledge of Riemann surface theory needed, as the Introduction covers the essentials. Additionally, it caters to specialists in geometry and mathematical physics who utilize Riemann surfaces in their research. This book is the first to reflect the progress made in this field over the last decade, featuring original results. The increasing applications of Riemann surfaces in evaluating concrete characteristics of models are highlighted, with many problem settings and computations motivated by practical applications in geometry and mathematical physics.

Since the time of surfaces -+ in differential Gauss, parametrized (x, y) P(x, y) have been described a frame attached to the moving geometry through TI(x, y) surface. One introduces the Gauss- which linear dif- Weingarten equations are, ferential equations = U = TIX T1, VT', !PY (1. for the and their condition frame, compatibility - = V + [U, V] 0, UY (1.2) which the Gauss-Codazzi For surfaces in three-dim- represents equations . a sional Euclidean the frame T1 lies in the usually or space, group SO(3) SU(2). On the other a of a non-linear in the form hand, representation equation (1.2) is the of the of of starting point theory integrable equations (theory solitons), which in mathematical in the 1960's appeared physics [NMPZ, AbS, CD, FT, More the differential for the coefficients of AbC]. exactly, partial equation (1.2) the matrices U and V is considered to be if these matrices can be integrable, extended to U V non-trivially a one-parameter family (x, y, A), (x, y, A) satisfying - = + U(A)y V(A). [U(A), V(A)] 0, (1-3) so that the differential is and original partial equation preserved.' . Usually U(A) V are rational functions of the which is called the (A) parameter A, spectral param- In soliton the eter is called the Lax . theory, representation (1.3) representation the Zakharov-Shabat or representation [ZS].