The new fields of mathematical visualization and mathematical geometry processing develop novel mathematical techniques to bridge the gap between the more classical fields of differential geometry (studying smooth curves and surfaces like the solutions to many variational problems) and discrete geometry (studying polyhedral surfaces, like the typical representations used in computers).

The new field of discrete differential geometry (Bobenko, Pinkall, Polthier, Sullivan) has many related strands of research, as demonstrated at the first Oberwolfach workshop in the area, held in March 2006. Among other things, it has been noted repeatedly that with proper choices of discretization for many smooth problems the discrete surfaces that arise are not merely approximations but share special properties of the smooth limits. For instance, for surfaces defined by integrable equations, the analogous discrete integrable systems shed great light on the essence of integrability. One can follow Whitney to develop discrete exterior calculus by requiring exact sequences at the discrete level. The related theory of discrete harmonic maps and discrete minimal surfaces can also be explained in terms of finite element methods and, therefore, directly links with numerical applications. Various techniques related to circle packings and circle patterns seem to give good notions of discrete conformal and holomorphic maps.

This geometric research, focusing on curves and surfaces in three-dimensional space, has many practical applications in addition to its theoretical interest. Typical systems for computer-aided design (CAD) rely heavily on spline surfaces for geometric modeling and design. In contrast, geometry acquisition systems like 3D scanners provide point sets and simplicial meshes not yet suitable for CAD systems. Here the labor-intensive reverse engineering process can be simplified by switching from spline representations to novel methods from discrete differential geometry. Similarly the requirements of computer graphics, both for real-time interactive games and for high-quality film rendering, provide a rich source of problems in geometry processing: how to efficiently manipulate digital representations of geometric structures.

The area in Berlin is represented by the DFG Research Group Polyhedral Surfaces at TU (Bobenko, Pinkall, Sullivan, Ziegler), by the MATHEON chairs “Mathematical Visualization” (Sullivan) at TU and “Mathematical Geometry Processing” (Polthier) at FU, and by the visualization group at ZIB (Deuflhard, Hege).