# Dupin Cyclides Passing through a Fixed Circle

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 3. Main Results

**Lemma**

**1.**

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Distinguishing Principal and Villarceau Circles

- Either represent only reducible cyclide surfaces: namely, a pair of touching spheres (where one of the spheres could be a plane or degenerates to a point); see Remark 4;
- Or generically represent cyclide surfaces with complex (rather than real) coefficients in (8); real surfaces appear only in lower-dimensional intersections with the two main families described in Theorems 3 and 4.

**Remark**

**3.**

**Remark**

**4.**

## 5. Proving Theorems 3 and 4

#### 5.1. Proof for Quartic Cyclides

- (i)
- ${u}_{1}\ne 0$, so that ${u}_{2}={u}_{3}=0$, and eventually ${u}_{4}=0$. The obtained ideal is reducible, with the prominent factor ${V}_{6}={u}_{1}^{2}({v}_{2}^{2}+{v}_{3}^{2})+4{v}_{4}^{2}$ after elimination of ${v}_{1}$. The localization ${V}_{7}\ne 0$ belongs to the principal circle component. The case ${V}_{6}=0$ simplifies to ${v}_{2}={v}_{3}={v}_{4}=2{v}_{1}-{u}_{1}^{2}=0$, and the cyclide degenerates to a double-sphere case.
- (ii)
- ${u}_{1}=0$, ${u}_{2}^{2}+{u}_{3}^{2}-2{u}_{4}=0$. Elimination of the variables ${u}_{1}$, ${u}_{2}$, ${u}_{3}$, ${u}_{4}$ gives us a principal ideal, and the generator factors with$${V}_{7}={({v}_{2}^{2}+{v}_{3}^{2})}^{3}+{({v}_{1}{v}_{2}^{2}+{v}_{1}{v}_{3}^{2}+2{v}_{4}^{2})}^{2}.$$The localization ${V}_{7}\ne 0$ belongs to the principal circle component. With ${V}_{7}=0$ we get ${v}_{2}={v}_{3}={v}_{4}=0$, and the resulting ideal contains the product ${({u}_{2}^{2}+{u}_{3}^{2}+2{v}_{1})}^{2}({u}_{2}^{2}+{u}_{3}^{2}+2{v}_{1}-4{r}^{2})$. Either of the factors leads to points on the principal circle component.

#### 5.2. Proof for Cubic Cyclides

## 6. Smooth Blending of Cyclides

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Remark**

**5.**

#### 6.1. Smooth Blending along Principal Circles

**Proposition**

**1.**

**Proof.**

**Remark**

**6.**

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**7.**

**Proposition**

**4.**

- (i)
- ${u}_{0}=1,\phantom{\rule{1.em}{0ex}}{u}_{2}^{2}+{u}_{3}^{2}=2{u}_{0}{u}_{4},\phantom{\rule{1.em}{0ex}}{v}_{1}=-{u}_{4},\phantom{\rule{1.em}{0ex}}{v}_{2}={v}_{3}={v}_{4}=0$;
- (ii)
- ${u}_{0}=1,\phantom{\rule{1.em}{0ex}}{u}_{2}={u}_{3}={v}_{2}={v}_{3}=0$, $\phantom{\rule{1.em}{0ex}}{u}_{4}=\frac{2{r}^{2}{u}_{1}(\lambda -{u}_{1})}{{\lambda}^{2}}$,$v}_{1}=\frac{{\lambda}^{2}{u}_{1}^{2}+4{r}^{2}{(\lambda -{u}_{1})}^{2}}{2{\lambda}^{2}$, $\phantom{\rule{1.em}{0ex}}{v}_{4}=\lambda {u}_{4}=\frac{2{r}^{2}{u}_{1}(\lambda -{u}_{1})}{\lambda}$.

**Proof.**

#### 6.2. Smooth Blending along Villarceau Circles

**Proposition**

**5.**

**Proof.**

## 7. The Möbius Invariant ${\mathit{J}}_{\mathbf{0}}$

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**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 $({u}_{0}:\dots :{u}_{4}:{v}_{1}:\dots :{v}_{4})$ are smoothly blended along the circle $\Gamma $ with $r=1$. The two cyclides on (e) are obtained from the parameter values $a=1$ and $a=1.8$. The two cyclides on (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 (${J}_{0}\phantom{\rule{-0.166667em}{0ex}}<0,r>R$); (

**b**) a horn torus (${J}_{0}=0$, $r=R$).

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**MDPI and ACS Style**

Menjanahary, J.M.; Vidunas, R.
Dupin Cyclides Passing through a Fixed Circle. *Mathematics* **2024**, *12*, 1505.
https://doi.org/10.3390/math12101505

**AMA Style**

Menjanahary JM, Vidunas R.
Dupin Cyclides Passing through a Fixed Circle. *Mathematics*. 2024; 12(10):1505.
https://doi.org/10.3390/math12101505

**Chicago/Turabian Style**

Menjanahary, Jean Michel, and Raimundas Vidunas.
2024. "Dupin Cyclides Passing through a Fixed Circle" *Mathematics* 12, no. 10: 1505.
https://doi.org/10.3390/math12101505