Channel surfaces in Lie sphere geometry

TL;DR

This paper explores the properties and transformations of channel surfaces within Lie sphere geometry, characterising them as $ Omega_{0}$-surfaces and examining their behaviour under various transformations and congruences.

Contribution

It introduces a new characterisation of channel surfaces as $ Omega_{0}$-surfaces and analyses their transformation behaviour and Ribaucour pairs using Dupin cyclide congruences.

Findings

Channel surfaces are characterised as $ Omega_{0}$-surfaces.

Transformation behaviour of channel surfaces under Lie sphere geometry is studied.

Ribaucour pairs of channel surfaces are characterised using Dupin cyclide congruences.

Abstract

We discuss channel surfaces in the context of Lie sphere geometry and characterise them as certain $\Omega_{0}$-surfaces. Since $\Omega_{0}$-surfaces possess a rich transformation theory, we study the behaviour of channel surfaces under these transformations. Furthermore, by using certain Dupin cyclide congruences, we characterise Ribaucour pairs of channel surfaces.