Ribaucour coordinates

TL;DR

This paper explores Ribaucour transformations for curves and submanifolds, introducing Ribaucour coordinates, dimension reduction results, and methods for smoothing and interpolating curvature line nets.

Contribution

It presents new results on Ribaucour transformations, including dimension reduction, the derivation of Ribaucour coordinates via Bianchi permutability, and techniques for smoothing and interpolating nets.

Findings

Dimension reduction of submanifolds using Ribaucour transformations

Introduction of Ribaucour coordinates based on Bianchi permutability

Methods for smoothing semi-discrete curvature line nets

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.