Dupin cyclidic systems geometrically revisited

TL;DR

This paper offers a geometric perspective on Dupin cyclidic systems, showing how they can be generated by evolving initial circles or cyclides, and relates Lamé families to parallel surfaces in different space forms.

Contribution

It introduces a new geometric approach to constructing Dupin cyclidic systems and connects Lamé families to parallel surfaces in space forms.

Findings

Dupin cyclidic systems can be obtained by evolving initial circles or cyclides.

Lamé families are characterized as parallel surfaces in space forms.

The geometric approach provides new insights into the structure of these systems.

Abstract

The induced metrics of Dupin cyclidic systems, that is, orthogonal coordinate systems with Dupin cyclides and spheres as coordinate surfaces, were provided by Darboux. Here we take a more geometric point of view and discuss how Dupin cyclides and Lam\'e families of Dupin cyclidic systems can be obtained by suitably evolving an initial circle or a Dupin cyclide, respectively. This approach reveals that those Lam\'e families are given by parallel surfaces in various space forms.