Ribaucour coordinates

Abstract

We discuss results for the Ribaucour transformation of curves or of higher dimensional smooth and discrete submanifolds. In particular, a result for the reduction of the ambient dimension of a submanifold is proved and the notion of Ribaucour coordinates is derived using a Bianchi permutability result. Further, we discuss smoothing of semi-discrete curvature line nets and an interpolation by Ribaucour transformations.

The Ribaucour families of discrete R-congruences

“…To conclude this section, we discuss some examples of special discrete R-congruences. This reveals relations to recent works on discrete Ribaucour coordinates [9] and discrete -surfaces [7]. Since we consider the discrete R-congruence as primary object, our approach sheds light on these situations from a different point of view.…”

“…In the last years, in a variety of works, discrete Ribaucour transformations between general discrete surfaces were discussed [9,20,25]. Furthermore, restricted transformations for particular classes of discrete surfaces as discrete (special) isothermic surfaces [4,11,12,22,23] and discrete O-surfaces [29] were investigated, to name just a few.…”

Discrete cyclic systems and circle congruences

“…Thus, the subspace γ 1 ⊕ γ 2 has to be a 4-dimensional space. [10,23] that any two cospherical circles are related by a Ribaucour transformation, that is, they envelop a common circle congruence. For two unparametrized circles on a sphere, there are at most two Ribaucour transformations that induce different Ribaucour correspondences between the point spheres of the two circles.…”

“…We remark that the flat connection of Prop 23 is not unique. However, if we fix a Möbius subgeometry by choosing a point sphere complex p such that the Ribaucour sphere congruence is non-degenerate in this Möbius subgeometry, we may fix homogeneous coordinates r : V → L so that r, p ≡ 1 and consider the discrete map (10) a :…”

Channel surfaces in Lie sphere geometry

“…Furthermore, the subclasses of channel linear Weingarten sur-faces (Hertrich-Jeromin et al 2015) and Willmore channel surfaces (Musso and Nicolodi 1999) were recently discussed. Moreover, a novel application of channel surfaces to semi-discrete curvature line nets was explored in Burstall et al (2017). Channel surfaces are also widely used in Computer Aided Geometric Design and the existence of a rational parametrisation was investigated in Peternell and Pottmann (1997).…”