Meshes for multilayer freeform structures

Abstract

The geometric problems which occur when realizing freefrom geometries as steel/glass structures can be effciently dealt with by the concept of parallel meshes [3]. These problems include the so-called node torsion problem, offsets of various types, and the constant beam height problem [2]. It turns out that a mesh has an offset mesh at constant face-face distance if and only if it has a parallel mesh which is circumscribed to the unit sphere. This property characterizes the so-called conical meshes and is relevant to multilayer constructions of constant width. A mesh has an offset at constant edge-edge distance if it has a parallel mesh which is edgewise circumscribed to the unit sphere. The class of such meshes is very interesting and is related to Koebe's theorem on the realization of graphs as edge graphs of polyhedra. It is relevant to the constant beam height problem. Geometry processing applications of the concept of parallel meshes are geometric modeling with such meshes, and approximating freeform surfaces with them. It turns out that parallelity of meshes has implications in discrete differential geometry as well. E.g., it turns out that discrete minimal surfaces defined in terms of face-based curvatures associated with a mesh and its parallel Gauss image mesh contain classes of discrete minimal surfaces which have been considered earlier. The discrete minimal surfaces, which occur in the context of this theory, have been investigated in the quad mesh case [3] and also in the hexagonal mesh case [1]. For such special surfaces, applications in architecture are not so obvious, but it is notable that they support equilibrium forces in their edges.