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
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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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