Channel surfaces in Lie sphere geometry

Abstract

We discuss channel surfaces in the context of Lie sphere geometry and characterise them as certain 0-surfaces. Since 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.

Discrete channel surfaces

“…In the realm of Lie sphere geometry, Blaschke gave another characterization of smooth channel surfaces using the Lie cyclides of a surface, special Dupin cyclides which make maximal-order contact with the surface along its curvature directions (cf. [1,17]): a surface in Lie sphere geometry is a channel surface if and only if (S4) its Lie cyclide congruence is constant along one of its curvature directions.…”

Dupin cyclidic systems geometrically revisited

“…yields the enveloped Ribaucour sphere congruence. As shown in [22], the Ribaucour sphere congruence then gives rise to two special congruences of Dupin cyclides, provided by the two families of (2, 1)-planes…”

“…We now turn to the Ribaucour pairs that are most relevant for this work, namely those that consist of two Dupin cyclides. Since both curvature sphere congruences of a Legendre map degenerate to sphere curves u → s 1 (u) and v → s 2 (v) if and only if the surface is a Dupin cyclide, the situation considerably simplifies (see also [22,Thm 4.5]): the Ribaucour cyclide congruences of such a Ribaucour pair also become 1-dimensional: This property guarantees that the cyclic systems associated to these types of Ribaucour pairs are DC-systems. We again fix a point sphere complex p and work in a Möbius subgeometry.…”

The Ribaucour families of discrete R-congruences

“…In [27,Definition 4.4], the existence of two special Dupin cyclide congruences for a smooth Ribaucour pair of Legendre maps was pointed out. We report on a similar construction in the discrete case:…”