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Dupin cyclides passing through a fixed circle

Jean Michel Menjanahary, Raimundas Vidunas

TL;DR

This work characterizes all Dupin cyclides passing through a fixed circle $\Gamma$ (taken as $x=0$, $y^2+z^2=r^2$) by restricting the coefficients of their implicit equations, exploiting the Möbius-invariant relationship between cyclides and tori. The main contribution is a complete algebraic description: the family of Darboux cyclides through $\Gamma$ splits into two real components, corresponding to $\Gamma$ being a Villarceau circle or a principal circle on the cyclide, with explicit polynomial and rank conditions for each. The authors prove these decompositions via elimination in a coefficient-ring setup, handling quartic and cubic cases separately, and use the results to address smooth blending of two cyclides along $\Gamma$ and to compute a Möbius invariant $J_0$ that distinguishes components and special degeneracies. These results yield practical tools for CAGD blending along circles and deepen the understanding of the Möbius-invariant structure of Dupin cyclides, including explicit formulas for $J_0$ and its behavior on the principal and Villarceau components.

Abstract

We derive algebraic equations on the coefficients of the implicit equation to characterize all Dupin cyclides passing through a fixed circle. The results are applied to solve the basic problems in CAGD about blending of Dupin cyclides along circles.

Dupin cyclides passing through a fixed circle

TL;DR

This work characterizes all Dupin cyclides passing through a fixed circle (taken as , ) by restricting the coefficients of their implicit equations, exploiting the Möbius-invariant relationship between cyclides and tori. The main contribution is a complete algebraic description: the family of Darboux cyclides through splits into two real components, corresponding to being a Villarceau circle or a principal circle on the cyclide, with explicit polynomial and rank conditions for each. The authors prove these decompositions via elimination in a coefficient-ring setup, handling quartic and cubic cases separately, and use the results to address smooth blending of two cyclides along and to compute a Möbius invariant that distinguishes components and special degeneracies. These results yield practical tools for CAGD blending along circles and deepen the understanding of the Möbius-invariant structure of Dupin cyclides, including explicit formulas for and its behavior on the principal and Villarceau components.

Abstract

We derive algebraic equations on the coefficients of the implicit equation to characterize all Dupin cyclides passing through a fixed circle. The results are applied to solve the basic problems in CAGD about blending of Dupin cyclides along circles.
Paper Structure (11 sections, 12 theorems, 56 equations, 3 figures)

This paper contains 11 sections, 12 theorems, 56 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main results
  4. Distinguishing principal and Villarceau circles
  5. Proving Theorems $\ref{['th:m3']}$ and $\ref{['th:m3p']}$
  6. Proof for quartic cyclides
  7. Proof for cubic cyclides
  8. Smooth blending of cyclides
  9. Smooth blending along principal circles
  10. Smooth blending along Villarceau circles
  11. The Möbius invariant $J_0$

Key Result

Theorem 2.1

The hypersurface in $\mathbb{R}^3$ defined by $(eq:gendarb1)$ is a Dupin cyclide only if the following $12$ polynomials vanish: and $\sigma_{12}K_1,\;\sigma_{12}L_1,\; \sigma_{12}M_1, \sigma_{13}K_1,\;\sigma_{13}L_1,\; \sigma_{13}M_1$.

Figures (3)

  • Figure 1: A smooth torus (a) and a smooth Dupin cyclide (b). The solid circles are principal circles and the dashed circles are Villarceau circles.
  • Figure 2: Two Dupin cyclide equations with different coefficient values $[u_0,\ldots,u_4;v_1,\ldots,v_4]$ are smoothly blended along the circle $\Gamma$ with $r=1$. The two cyclides in (e) are obtained from the parameter values $a=1$ and $a=1.8$. The two cases at (f) are obtained from the parameter values $t=0$ and $t=0.4$.
  • Figure 3: A cutaway view of singular toruses: (a) a spindle torus ($J_0\!<0,r>R$); (b) a horn torus ($J_0=0$, $r=R$).

Theorems & Definitions (20)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 4.1
  • Remark 4.2
  • Lemma 6.1
  • ...and 10 more