# Darboux cyclides

## Overview

We show animations of Darboux cyclides together with some of their inversions.

A surface X is of **celestial type** (c,d,n) if

- the surface X contains c circles through almost each point,
- the degree of X is equal to d, and
- X is embedded into n-dimensional space and not contained in a hyperplane section.

The real structure and singular locus of X can be characterized by a pair of Dynkin types. The underlined Dynkin types correspond to real isolated singularities. See here for more information. For the names of the cyclides we use the following abbreviations:

E = elliptic/ellipsoid, | C = circular, |

H = hyperbolic/hyperboloid, | P = parabolic/paraboloid, |

O = cone, | Y = cylinder. |

1 = of one sheet, | 2 = of two sheets, |

A Darboux cyclide is either one of the following:

Blum cyclide | EO cyclide |

Perseus cyclide | CO cyclide |

ring cyclide | EE cyclide |

EH1 cyclide | EH2 cyclide |

CH1 cyclide | EP cyclide |

HP cyclide | S1 cyclide |

EY cyclide | S2 cyclide |

CY cyclide |

The images were made using Surfex.

## Blum cyclide

Celestial type: (6,4,3)

Real structure: 2A1

Singular locus: A0

## Perseus cyclide

Celestial type: (5,4,3)

Real structure: 2A1

Singular locus: 2A1

## ring cyclide

Celestial type: (4,4,3)

Real structure: 2A1

Singular locus: 4A1

## EH1 cyclide

Celestial type: (4,4,3)

Real structure: 2A1

Singular locus: __A1__

## CH1 cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: __A1__+2A1

## HP cyclide

Celestial type: (2,4,3)

Real structure: 2A1

Singular locus: __A2__

## EY cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: __A3__

## CY cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: __A3__+2A1

a.k.a. horn cyclide

## EO cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: 2__A1__

## CO cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: 2__A1__+2A1

a.k.a. spindle cyclide

## EE cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: __A1__

## EH2 cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: __A1__

## EP cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: __A2__

## S1 cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: A0

## S2 cyclide

Celestial type: (2,4,3)

Real structure: D4

Singular locus: A0