xiv Introduction surfaces, which turned out to be extremely important in geometry processing where it found numerous applications, e.g., Desbrun-Meyer-Alliez (2002), Botsch-Kobbelt (2004), to name but two. Another example is the theory of discrete minimal surfaces by Bobenko-Hoffmann-Springborn (2006), which turned out to have striking convergence properties: these discrete surfaces approximate their smooth analogs with all derivatives. A straightforward way to discretize differential geometry would be to take its analytic description in terms of differential equations and to apply standard methods of numerical analysis. Such a discretization makes smooth problems amenable to numerical methods. Discrete differential geometry does not proceed in this way. Its main message is: Discretize the whole theory, not just the equations. This means that one should develop a discrete theory which respects fundamental aspects of the smooth one which of the properties are to be taken into account is a nontrivial problem. The discrete geometric the- ory turns out to be as rich as its smooth counterpart, if not even richer. In particular, there are many famous existence theorems at the core of the clas- sical theory. Proper discretizations open a way to make them constructive. For now, the statement about the richness of discrete differential geometry might seem exaggerated, as the number of supporting examples is restricted (although steadily growing). However, one should not forget that we are at the beginning of the development of this discipline, while classical differ- ential geometry has been developed for centuries by the most outstanding mathematicians. As soon as one takes advantage of the apparatus of differential equations to describe geometry, one naturally deals with parametrizations. There is a part of classical differential geometry dealing with parametrized surfaces, coordinate systems and their transformations, which is the content of the fundamental treatises by Darboux (1914-27) and Bianchi (1923). Nowadays one associates this part of differential geometry with the theory of integrable systems see Fordy-Wood (1994), Rogers-Schief (2002). Recent progress in discrete differential geometry has led not only to the discretization of a large body of classical results, but also, somewhat unexpectedly, to a better understanding of some fundamental structures at the very basis of the classical differential geometry and of the theory of integrable systems. It is the aim of this book to provide a systematic presentation of current achievements in this field. Returning to the analytic description of geometric objects, it is not sur- prising that remarkable discretizations yield remarkable discrete equations.

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