Discretizations of Surfaces with Constant Ratio of Principal Curvatures

Abstract

Motivated by applications in architecture, we study surfaces with a constant ratio of principal curvatures. These surfaces are a natural generalization of minimal surfaces, and can be constructed by applying a Christoffel-type transformation to appropriate spherical curvature line parametrizations, both in the smooth setting and in a discretization with principal nets. We link this Christoffel-type transformation to the discrete curvature theory for parallel meshes and characterize nets that admit these transformations. In the case of negative curvature, we also present a discretization of asymptotic nets. This case is suitable for design and computation, and forms the basis for a special type of architectural support structures, which can be built by bending flat rectangular strips of inextensible material, such as sheet metal.

Caravel meshes: A new geometrical strategy to rationalize curved envelopes

“…That is, the surface is both isothermic and close to a Weingarten surface 2√3 1 + 2 = 0 in which the ratio of principal curvature is a constant. The generation and the interesting design potential of this family of surfaces was recently studied in (Jimenez et al 2019).…”