Dupin Cyclides as a Subspace of Darboux Cyclides

TL;DR

This paper characterizes Dupin cyclides as a special subset of Darboux cyclides by deriving algebraic conditions for their recognition and classifies all real surfaces and degenerations within this class.

Contribution

It provides explicit algebraic criteria to identify Dupin cyclides among Darboux cyclides and classifies all related real surfaces and degenerations.

Findings

Algebraic conditions for recognizing Dupin cyclides derived

Complete intersection sets of equations defined for Dupin cyclides

Classification of all real surfaces and degenerations related to Dupin cyclides

Abstract

Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and to construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather general surfaces in $\mathbb R^3$ of degree 3 or 4. This article derives the algebraic conditions for recognition of Dupin cyclides among the general implicit form of Darboux cyclides. We aim at practicable sets of algebraic equations on the coefficients of the implicit equation, each such set defining a complete intersection (of codimension 4) locally. Additionally, the article classifies all real surfaces and lower dimensional degenerations defined by the implicit equation for Dupin cyclides.