2006/10/13 by Alexander I. Bobenko, Yuri B. Suris · 2 citations
Mathematics · Physics and Astronomy · #Combinatorics #Diagonal #Geometric and Algebraic Topology #Geometry #Laguerre polynomials #Mathematical analysis #Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Plane (geometry) #Projective plane #Pure mathematics #Vertex (graph theory) #math.DG
paper · pdf · doi:10.1098/rspa.2007.1902
published as Proc. Royal Soc. A, 2007, 463, p. 3171-3193 · 29 pages, 8 figures
arxiv created 2006/10/13 · openalex publication_date 2007/09/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We give an elaborated treatment of discrete isothermic surfaces and their analogues in different geometries (projective, Möbius, Laguerre and Lie). We find the core of the theory to be a novel characterization of discrete isothermic nets as Moutard nets. The latter are characterized by the existence of representatives in the space of homogeneous coordinates satisfying the discrete Moutard equation. Moutard nets admit also a projective geometric characterization as nets with planar faces with a five-point property: a vertex and its four diagonal neighbours span a three-dimensional space. Restricting the projective theory to quadrics, we obtain Moutard nets in sphere geometries. In particular, Moutard nets in Möbius geometry are shown to coincide with discrete isothermic nets. The five-point property, in this particular case, states that a vertex and its four diagonal neighbours lie on a common sphere, which is a novel characterization of discrete isothermic surfaces. Discrete Laguerre isothermic surfaces are defined through the corresponding five-plane property, which requires that a plane and its four diagonal neighbours share a common touching sphere. Equivalently, Laguerre isothermic surfaces are characterized by having an isothermic Gauss map. S-isothermic surfaces as an instance of Moutard nets in Lie geometry are also discussed.