Discrete differential geometry and Klein's Erlangen program

Discrete differential geometry (DDG) aims at the development of discrete equivalents of the concepts and methods of classical differential geometry. The latter appears as a limit of a refinement of the discretization. Current interest in DDG derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, and architectural geometry. DDG is based on the concept of structure-preserving discretization - a discrete theory that respects the fundamental properties of smooth geometry. A discretization principle originating from the Klein Erlangen program plays a crucial role in establishing these discretizations. Its main message is, "discretize your problem within the geometry to which it belongs".

We demonstrate this principle on numerous examples, including conformal and curvature line parametrized surfaces, as well as the Willmore energy functional. We also show how structure preserving discretization helped to solve the classical Bonnet and Berger problems of the global surface theory.