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.
Highlights
Channel surfaces, that is, envelopes of one-parameter families of spheres, have been intensively studied for many years
Since 0-surfaces possess a rich transformation theory, we study the behaviour of channel surfaces under these transformations
In Bernstein (2001), Hertrich-Jeromin (2003), HertrichJeromin et al (2001), Jensen et al (2016) and Musso and Nicolodi (1999, 2002) channel surfaces were studied in the context of Möbius geometry and in Musso and Nicolodi (1995) and Peternell and Pottmann (1998) they were given a Laguerre geometric treatment
Summary
That is, envelopes of one-parameter families of spheres, have been intensively studied for many years These surfaces are a classical notion (e.g., Blaschke 1929; Lie 1872; Monge 1850), they are a subject of interest in recent research. In Musso and Nicolodi (2006) it was shown that 0surfaces are deformable surfaces in Lie sphere geometry This induces a transformation of 0-surfaces called the Calapso transformation. Given a pair of sphere curves, we construct two 1-parameter families of Dupin cyclides whose coincidence determines when the envelopes of the sphere curves form a Ribaucour pair (with corresponding circular curvature lines). Any Ribaucour pair of channel surfaces (with corresponding circular curvature lines) can be arranged as a Lie-Darboux pair. We recover a result of Burstall and Hertrich-Jeromin (2006), showing how the classical notion of Ribaucour transforms of curves is related to the Ribaucour transforms of Legendre immersions parametrising these curves
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