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.
Highlights
From a perspective of higher geometries, channel surfaces form a class of simple surfaces, as they can be considered as curves in a suitable space of geometric objects, e.g. a space of spheres or a space of Dupin cyclides
This leads to another characterization of discrete Legendre maps in terms of face-cyclides: Proposition 1.4 A congruence of Dupin cyclides described by pairs of orthogonal (2, 1)
In this realm of Ribaucour transformations, we investigate the geometry of two curvature lines of a discrete channel surface (Fig. 5)
Summary
From a perspective of higher geometries, channel surfaces form a class of simple surfaces, as they can be considered as curves in a suitable space of geometric objects, e.g. a space of spheres or a space of Dupin cyclides. As such, they are well suited for applications in engineering or computer aided geometric design, as they can be described in a simpler way than more general surfaces. We find that a geometric object may not be defined at just one type of cells in the underlying cell complex, but be distributed across cells of different dimensions This approach is based on the analysis of enveloped sphere congruences, and on enveloped congruences of Dupin cyclides, cf [4]. The approach we propose carries directly over to a semi-discrete setting
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