Discrete channel surfaces

TL;DR

This paper introduces a new discrete model of channel surfaces within Lie sphere geometry, enabling reconstruction of these surfaces from discrete data and extending classical theorems to the discrete setting.

Contribution

It defines discrete channel surfaces in Lie sphere geometry and establishes a discrete version of Vessiot's Theorem for isothermic discrete channel surfaces.

Findings

Defined discrete channel surfaces reflecting properties of smooth ones

Provided methods to reconstruct surfaces from vertex, edge, or face data

Proved a discrete analogue of Vessiot's Theorem for isothermic cases

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.