Discrete channel surfaces

Abstract

We present a definition of discrete channel surfaces in Lie sphere geometry, which reflects several properties for smooth channel surfaces. Various sets of data, defined at vertices, on edges or on faces, are associated with a discrete channel surface that may be used to reconstruct the underlying particular discrete Legendre map. As an application we investigate isothermic discrete channel surfaces and prove a discrete version of Vessiot's Theorem.

The Ribaucour families of discrete R-congruences

“…To prove the second property of the discrete R-congruence, let us consider a (1)-coordinate ribbon and denote the constant curvature spheres of f andf along the two boundary (1)coordinate lines by s i , s i+1 ,ŝ i andŝ i+1 , respectively. Furthermore, by Hertrich-Jeromin et al [24,Proposition 2.4], the other family of curvature spheres of f along this (1)-coordinate ribbon lies in a (2, 1)-plane D i .…”

“…For any face of a discrete Legendre map there exists a 1-parameter family of face-cyclides, Dupin cyclides that share the four curvature spheres assigned to a face with the discrete Legendre map (cf. [3,24]). Note that any face-cyclide has four distinguished curvature lines, namely the curvature lines that join two adjacent contact elements of the discrete Legendre map and lie on the corresponding curvature sphere.…”

“…In this section, we will draw attention to the Ribaucour transformation of discrete channel surfaces as discussed in [24] and the wider class of discrete Legendre maps with a family of spherical curvature lines.…”

“…We recall that a discrete Legendre map is a discrete channel surface in the sense of Hertrich-Jeromin et al [24], if it admits a face-cyclide congruence which is constant along one family of coordinate ribbons.…”

“…Geometrically, those discrete R-congruences are provided by circular nets where the point spheres of one family of coordinate lines lie on circles such that two adjacent ones are related by a Möbius inversion. Then, by Corollary 5.5, there exists a 1-parameter choice of contact elements such that the constructed principal net provides a discrete channel surface, that is, the principal net has indeed a constant curvature sphere along each coordinate line of one family (for details see also [24]).…”