On the unification of classical and novel integrable surfaces: II. Difference geometry

Abstract

A novel class of discrete integrable surfaces is recorded. This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, `linear' Weingarten, Guichard and Petot surfaces. Moreover, natural discrete analogues of the Gaussian and mean curvatures for surfaces parametrized in terms of curvature coordinates are used to define surfaces of constant discrete Gaussian and mean curvatures and discrete minimal surfaces. Remarkably, these turn out to be proto-typical examples of discrete O surfaces. It is demonstrated that the construction of a Backlund transformation for discrete O surfaces leads in a natural manner to an associated parameter-dependent linear representation. Canonical discretizations of the classical pseudosphere and breather pseudospherical surfaces are generated. Connections with pioneering work by Bobenko and Pinkall are established.