Introduction What is discrete differential geometry. A new field of discrete dif- ferential geometry is presently emerging on the border between differential and discrete geometry see, for instance, the recent book Bobenko-Schr¨oder- Sullivan-Ziegler (2008). Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), dis- crete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. The latter appears as a limit of refinement of the discretization. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields: see the lecture course on discrete differential geometry in computer graphics by Desbrun-Grinspun-Schr¨ oder (2005), the recent book on architectural geometry by Pottmann-Asperl-Hofer-Kilian (2007), and the mathematical video on polyhedral meshes and their role in geometry, nu- merics and computer graphics by Janzen-Polthier (2007). For a given smooth geometry one can suggest many different discretiza- tions with the same continuous limit. Which is the best one? From the theo- retical point of view, one would strive to preserve fundamental properties of the smooth theory. For applications the requirements of a good discretiza- tion are different: one aims at the best approximation of a smooth shape, on the one hand, and at on the other hand, its representation by a discrete shape with as few elements as possible. Although these criteria are different, it turns out that intelligent theoretical discretizations are distinguished also by their good performance in applications. We mention here as an example the discrete Laplace operator on simplicial surfaces (“cotan formula”) intro- duced by Pinkall-Polthier (1993) in their investigation of discrete minimal xiii
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