Classification of Dupin Cyclidic Cubes by Their Singularities

TL;DR

This paper classifies Dupin cyclidic cubes, a type of triple orthogonal coordinate system with circular or linear lines, focusing on their singularities and M"obius equivalence in Euclidean space.

Contribution

It introduces a classification of Dupin cyclidic cubes based on their singularities, extending the understanding of these coordinate systems beyond previous bilinear parametrizations.

Findings

Classification of Dupin cyclidic cubes up to M"obius equivalence

Analysis of singularities in these coordinate systems

Generalization of bilinear parametrizations of Dupin cyclides

Abstract

Triple orthogonal coordinate systems having coordinate lines as circles or straight lines are considered. Technically, they are represented by trilinear rational quaternionic maps and are called Dupin cyclidic cubes, naturally generalizing the bilinear rational quaternionic parametrizations of principal patches of Dupin cyclides. Dupin cyclidic cubes and their singularities are studied and classified up to M\"obius equivalency in Euclidean space.