Semidiscrete Surfaces

Semidiscrete surfaces comprise the entire spectrum of smooth mappings of both integer and real variables, with the purely discrete and purely smooth surfaces as extremal cases. They can be seen in an effective manner as limit objects of a discrete master theory which remarkably involves integrable systems and incidence geometry, and they are classically employed in the transformation theory of surfaces. The lowest-dimensional case of one discrete and one continuous variable attracted attention in its own right when it occurred in problems related to freeform architecture. Such 2D semidiscrete surfaces deserve further study and bear a great application potential: Within the present subproject we propose to study the semidiscrete manifestations of minimal and cmc surfaces, transformations, flexions, and their applications. Semidiscrete surfaces as we understand them made their appearance in an applied paper on approximation of freeform shapes by a union of single-curved smooth panels - from a higher viewpoint this is conjugate semidiscrete surfaces and their circular and conical reductions. Also the A-surfaces, where parameter lines are asymptotic in a certain well-defined semidiscrete sense, have turned up in real-world problems. The very successful principle of a discrete master theory (see the recent textbook by Bobenko and Suris) implies that semidiscrete surfaces are nothing but a limit of the well studied discrete case. In fact the latter is a very good guide but semidiscrete surfaces do exhibit geometric properties specific to the presence of both discrete and smooth parameters. We have already investigated the A-surfaces and the important special case of K-surfaces which exhibit constant Gaussian curvature: while the moving frames associated with K-surfaces and the resulting sine-Gordon and Hirota equations are analogous to the discrete case, an interesting difference is that the value K may be defined already for A-surfaces. This fact converts the usual definition of discrete K-surfaces by means of the Chebychev property into a theorem. Other work in this area concerns circular surfaces and their conformal mappings, and duality of minimal surfaces. The prominent incidence-geometric characterizations of discrete surface are partly retained in the semidiscrete case. The goal of this project is to extend our knowledge of semidiscrete surfaces in the following areas. *) Curvatures, minimal surfaces and cmc surfaces: The recent curvature theory based on edgewise parallel Gauss images nicely extends to semidiscrete surfaces. Interesting are cases where the Gauss image is uniquely determined by the original surface, such as for conical surfaces. *) Transformations: Surprisingly this classical concept already appeared in applications (in connection with multilayer structures in freeform architecture). Here appropriate partial limits of discrete integrable systems come into play. *) Flexibility: This wide and presumably difficult area is one where the respective behaviours of smooth and discrete surfaces differ. Unsolved problems include the semidiscrete analogue of Bianchi`s rigidity condition, and flexions and infinitesimal flexions of conjugate surfaces considered as metric spaces. Here we are guided by Pogorelov`s work on embeddings, at least in the convex case. *) Applications. The optimization problems regarding discrete and semidiscrete surfaces encountered in the past exhibit unfriendly numerics and confusing convergence behaviour if not properly initialized. Geometric knowledge, and especially the analogy between different categories of surfaces turned out to be essential here.

This project is part of the greater SFB-Transregio programme Discretization in Geometry and Dynamics which is funded by Deutsche Forschungsgemeinschaft and the Austrian Science Fund. Its topic is semidiscrete surfaces, which means parametric surfaces with both discrete and continuous parameters. Such mixed objects have been present ever since the 19th century transformation theory of smooth surfaces. When both discrete and continuous surfaces, and their transformations, were subsumed under a discrete master theory of integrable systems, mixed objects occur naturally. Interestingly, even the 2-dimensional case of surfaces with one continuous and one discrete parameter has a rich structure and deserves special attention. In the course of this project we have studied entities of classical differential geometry and ways of extending them to discrete and semidiscrete surfaces in natural ways. Here we could draw on the rapidly developing body of recent work on discrete differential geometry. Topics were the Laplacian on surfaces, and a curvature theory based on the Steiner formula. A nice result is the construction of surfaces of constant mean curvature with their associated families. As to applications, semidiscrete objects occur in freeform architectural skins, and there are some instances where mathematical theory and the practice of building construction are mutually beneficial. For example we were able to contribute to the geometry of shading and lighting systems, and the analysis and design of curved support structures. We were also working on numerical differential geometry and geometry processing, a particular highlight being a computational approach to the interactive design of developable surfaces and curved-folding surfaces.