Discrete surfaces for architectural design

Geometric problems originating in architecture can lead to interesting research and results in geometry processing, computer aided geometric design, and discrete differential geometry. In this article we survey this development and consider an important problem of this kind: Discrete surfaces (meshes) which admit a multi-layered geometric support structure. It turns out that such meshes can be elegantly studied via the concept of parallel mesh. Discrete versions of the network of principal curvature lines turn out to be parallel to approximately spherical meshes. Both circular meshes and the conical meshes considered only recently are instances of this meta-theorem. We dicuss properties and interrelations of circular and conical meshes, and also their connections to meshes in static equilibrium and discrete minimal surfaces. We conclude with a list of research problems in geometry which are related to architectural design.