Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dupin Cyclides as a Subspace of Darboux Cyclides

Jean Michel Menjanahary, Raimundas Vidunas

TL;DR

The paper addresses the problem of recognizing Dupin cyclides as a distinguished subspace of the general Darboux cyclides by deriving explicit algebraic conditions on the implicit coefficients in $\mathbb{P}^{13}$, separating quartic and cubic cases into ${\mathcal{D}}_4$ and ${\mathcal{D}}_3$ with codimension 4 inside the ambient space.Its approach combines canonical forms, orthogonal (and Möbius) transformations, and computational algebra (Gröbner bases) to produce practical, complete-intersection descriptions for the quartic and cubic subspaces, plus a real-case classification.Key contributions include explicit polynomial criteria defining ${\mathcal{D}}_4^*$ and ${\mathcal{D}}_3$, a stratification into open components, a discussion of limits between quartic and cubic cyclides, and the Möbius-invariant $J_0$ that classifies Dupin cyclides up to Möbius transformations.The results enable robust recognition and classification of Dupin cyclides in geometric design and architecture, provide a structured view of the full Dupin cyclide space, and connect algebraic descriptors to geometric features and degenerations.

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.

Dupin Cyclides as a Subspace of Darboux Cyclides

TL;DR

The paper addresses the problem of recognizing Dupin cyclides as a distinguished subspace of the general Darboux cyclides by deriving explicit algebraic conditions on the implicit coefficients in $\mathbb{P}^{13}$, separating quartic and cubic cases into ${\mathcal{D}}_4$ and ${\mathcal{D}}_3$ with codimension 4 inside the ambient space.Its approach combines canonical forms, orthogonal (and Möbius) transformations, and computational algebra (Gröbner bases) to produce practical, complete-intersection descriptions for the quartic and cubic subspaces, plus a real-case classification.Key contributions include explicit polynomial criteria defining ${\mathcal{D}}_4^*$ and ${\mathcal{D}}_3$, a stratification into open components, a discussion of limits between quartic and cubic cyclides, and the Möbius-invariant $J_0$ that classifies Dupin cyclides up to Möbius transformations.The results enable robust recognition and classification of Dupin cyclides in geometric design and architecture, provide a structured view of the full Dupin cyclide space, and connect algebraic descriptors to geometric features and degenerations.

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 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.
Paper Structure (19 sections, 8 theorems, 73 equations, 2 figures)

This paper contains 19 sections, 8 theorems, 73 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The main results
  3. Recognition of quartic Dupin cyclides
  4. Recognition of cubic Dupin cyclides
  5. Quartic Dupin cyclides
  6. From the canonical form
  7. Applying orthogonal transformations
  8. Elimination of the coefficients of the canonical form
  9. Proof of Theorem \ref{['th:m1']}
  10. Cubic Dupin cyclides
  11. Applying orthogonal transformations
  12. Proof of Theorem 2.3
  13. The whole space of Dupin cyclides
  14. Cubic cyclides as limits of quartic cyclides
  15. Dimension, degree, Hilbert series
  16. ...and 4 more sections

Key Result

Theorem 2.1

The hypersurface in $\mathbb{R}^3$ defined by $(eq:mainForm)$ is a Dupin cyclide if and only if one of the following cases holds:

Figures (2)

  • Figure 1: The inclusion diagram for the varieties of Dupin cyclides embedded in the spaces of Darboux cyclides.
  • Figure 2: (a) Classification of real points on quartic cyclides in the canonical form (\ref{['eq:dupin']}). (b) Classification of real points on toruses (\ref{['eq:torus']}). (c) Legend.

Theorems & Definitions (21)

  • Theorem 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem 2.4
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 11 more