Discrete $Ω$-nets and Guichard nets via discrete Koenigs nets

Abstract

We provide a convincing discretisation of Demoulin's Ω-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.

Notes on flat fronts in hyperbolic space

“…To set the scene we fix some notations, cf [6] or [7]: we shall consider maps that "live" on cells of some dimension of a quadrilateral cell complex,…”

“…2 ), where Σ 2 0 denotes the set of vertices, Σ 2 1 the set of (oriented) edges, and Σ 2 2 the set of (oriented) quadrilateral faces; the elements of Σ 2 1 and Σ 2 2 will usually be denoted by their vertices, i.e., (ij) ∈ Σ 2 1 denotes an edge from i ∈ Σ 2 0 to j ∈ Σ 2 0 , and (ijkl) ∈ Σ 2 2 denotes an oriented face with vertices i, j, k, l ∈ Σ 2 0 and edges (ij), (jk), (kl), (li) ∈ Σ 2 1 . We will generally assume Σ 2 to be simply connected, that is, any two points i, j ∈ Σ 2 0 can be connected by an (edge-)path and any closed (edge-)path is null-homotopic, via "face-flips" and via dropping single edge "return trips", cf [7,Sect 2.3].…”

Weierstrass-type representations

“…Thus, every Λ ∈ Z aff is identified with a contact element. In accordance with [7] we pose the following definition.…”

“…Discrete L-isothermic surfaces. In Laguerre geometry a discrete isothermic sphere congruence is a discrete surface x : Z 2 → R 3,1 that possesses a Christoffel dual x * : Z 2 → R 3,1 , which is characterised by (see [7,Thm 4.11])…”

“…discrete L-isothermic surfaces originally described in [8] (see also [7]), which we will develop in the realm of Laguerre geometry. In particular, we will obtain formulas for the Calapso transformation of discrete L-isothermic surfaces in the Hermitian model of R 3,1 .…”