Surfaces with piecewise linear support functions over spherical triangulations

Given a smooth surface patch we construct an approximating piecewise linear structure. More precisely, we produce a mesh for which virtually all vertices have valency three. We present two methods for the construction of meshes whose facets are tangent to the original surface. These two methods can deal with elliptic and hyperbolic surfaces, respectively. In order to describe and to derive the construction, which is based on a projective duality, we use the so-called support function representation of the surface and of the mesh, where the latter one has a piecewise linear support function.