2024/10/11 by Alexander I. Bobenko, Tim Hoffmann, Bobenko, Alexander I. +3
Engineering · Mathematics · #52C26 #53A10 (primary) #53C42 (secondary) #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2410.08915
openalex publication_date 2024/10/11 · openalex created_date 2024/10/16 · openalex updated_date 2026/07/28
We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles. These discrete cmc surfaces can be constructed from orthogonal ring patterns in the two-sphere and the hyperbolic plane. We present a variational principle that allows us to solve boundary value problems and to construct discrete analogues of some classical cmc surfaces. The data used for the construction is purely combinatorial - the combinatorics of the curvature line pattern. In the limit of orthogonal circle patterns we recover the theory of discrete minimal surfaces associated to Koebe polyhedra all edges of which touch a sphere. These are generalized to two-sphere Koebe nets, i.e., nets with planar quadrilateral faces and edges that alternately touch two concentric spheres.