Discrete geometric structures motivated by applications

Contemporary architecture of large scale buildings makes it necessary to segment surfaces which will be realized as roofs, walls or facades into smaller parts, the so called panels. Thus the application implies the highly relevant question of how to subdivide a surface in a meaningful way which leads to challenging basic research questions within the field of geometry and in particular discrete differential geometry. While the building industry, architects, and engineers are mainly interested in design aspects, realizability and statics, we will focus on studying the underlying mathematical research questions which have been motivated by architecture. Our research project will be carried out in close collaboration between TU Berlin and TU Wien within the framework of the CRC Transregio Discretization in Geometry and Dynamics. Our main focus lies on discretizing different parametrizations of surfaces. One topic therein is the study of curves with constant normal curvature which is a special case of parametrizations which are in a certain way diagonal to the classical parametrizations. An important basic principle hereby as well as for the following topics is not to simply discretize the characterizing differential equations but rather to discretize the methods and thereby the theory in which our topics are imbedded. Furthermore, we will analyze circle patterns, which already have appeared in many outstanding scientific publications as particularly relevant objects of discrete differential geometry. We particularly plan to generalize these circle patterns to circular ring patterns, for which we expect interesting relations to integrable systems and hyperbolic geometry. Another topic which is directly attached to the panelization problem is to direct our attention to the faces of a discretized net. Many discretizations of nets in sphere geometries ignore the geometry of the faces and consider nets merely as set or sequence of vertices. We will study nets where their faces are made from sphere geometric objects and will derive new methods therewith. Finally it is also our goal to generalize subdivision algorithms, which have appeared in computer graphics and geometric modelling as linear schemes over the last decades, into our nonlinear world of sphere geometries. First results in this direction appear to provide promising approaches for this nonlinear research problem.