Darboux cyclides

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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 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 cyclideCO 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

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Perseus cyclide

Celestial type: (5,4,3)

Real structure: 2A1

Singular locus: 2A1

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ring cyclide

Celestial type: (4,4,3)

Real structure: 2A1

Singular locus: 4A1

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EH1 cyclide

Celestial type: (4,4,3)

Real structure: 2A1

Singular locus: A1

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CH1 cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: A1+2A1

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HP cyclide

Celestial type: (2,4,3)

Real structure: 2A1

Singular locus: A2

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EY cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: A3

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CY cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: A3+2A1

a.k.a. horn cyclide

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EO cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: 2A1

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CO cyclide

Celestial type: (3,4,3)

Real structure: 2A1

Singular locus: 2A1+2A1

a.k.a. spindle cyclide

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EE cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: A1

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EH2 cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: A1

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EP cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: A2

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S1 cyclide

Celestial type: (2,4,3)

Real structure: 3A1

Singular locus: A0

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S2 cyclide

Celestial type: (2,4,3)

Real structure: D4

Singular locus: A0

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