# Surfaces that contain two circles through each point

## Table of contents

See here for animations of Darboux cyclides.

## Celestial surfaces

A
**celestial surface**
contains at least two real circles through a general point.
We consider lines as circles with infinite radius.

In 1669, Sir Christopher Wren wrote that a one-sheeted hyperboloid contains two lines through each point. Such surfaces are of interest to architects.

Kobe Port Tower | Shukov Tower |

Strasbourg Cathedral | ring torus |

In 1848, Yvon Villarceau wrote that the ring torus contains four circles through a general point. The diagonal circles are called Villarceau circles. A staircase in the Strasbourg Cathedral has sculptors with Villarceau circles and the construction of the cathedral took place between 1176 and 1439. Richard Blum constructed in 1980 celestial surfaces that contain more than four circles through each point.

Blum cyclide | ||

Perseus cyclide |

Celestial surfaces are of recent interest in geometric modeling [1].

## Celestial surfaces in higher dimensional space

There are celestial surfaces that can not be embedded into 3-dimensional space.

We say that a celestial surface X is of 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.

We showed in [2] that a celestial surface is of type either

- (2,8,7),
- (2,8,6),
- (2,8,5), (3,6,5), (2,6,5),
- (2,8,4), (2,6,4), (3,6,4), (∞,4,4),
- (2,8,3), (6,4,3), (5,4,3), (4,4,3), (3,4,3), (2,4,3) or
- (∞,2,2).

Celestial surfaces such that (d,n) equals (4,3) are called
**Darboux cyclides**.
Projections of celestial surfaces with n≥4 are covered by ellipses instead of circles!

(3,6,5) | (2,6,4) | (∞,4,4) |

## Dupin cyclides

A Dupin cyclide can be defined as a celestial surface that is the orbit of a point under a 2-dimensional subgroup of the Möbius group. We showed in [3] that Dupin cyclides are of type either

- (2,8,7), (2,8,5), (3,6,5), (∞,4,4), (4,4,3), (2,4,3) or (∞,2,2).

János Kollár showed in [4] that celestial surfaces of type (∞,4,4) are unique up to Möbius equivalence.

ring torus | spindle torus | horn torus |

(4,4,3) | (2,4,3) | (2,4,3) |

## Translational celestial surfaces

A Bohemian dome
is a translation of a circle along another circle in 3-dimensional Euclidean space such that each circle is parallel.
Such **Euclidean translational celestials** are the pointwise sum of circles in Euclidean 3-space.
**Clifford translational celestials** are the counterpart in elliptic geometry
and are realized as the pointwise Hamiltonian product of circles in the unit-quaternions.

Mikhail Skopenkov and Rimvydas Krasauskas showed in [5] that a celestial surface in 3-dimensional space is either Euclidean translational, Clifford translational or a Darboux cyclide.

We classified in [6, Theorem 1 and 2], the translational celestial surfaces in terms of their singular loci; the figures in the table below show at least one representative for each equivalence class.

Clifford translations of great circles along little circles are, up to homeomorphism, equal to a normal form that is shaped like one of the four images in the 5th block below (see [6, Corollary 4]). The complete classification, up to homeomorphism, of celestial surfaces that do not contain great circles remains an open problem (see [6, Remark 1]).

Euclidean translations of lines along lines | |

Euclidean translations of lines along circles | |

Euclidean translations of circles along circles | |

Clifford translations of great circles along great circles | |

Clifford translations of great circles along little circles | |

Clifford translations of little circles along little circles | |

Animations of Clifford translations of great circles along little circles |

It is natural to embed celestial surfaces in spheres. See the following websites to get some feeling for the geometry of the 3-sphere:

- Moebius Transformations Revealed (for the two-sphere)
- Dimensions: A walk through mathematics (Chapter 7)
- Space Symmetry Structure: 4-dimensional rotations (Daniel Piker)

## References

[1] |
Darboux cyclides and webs from circles
H. Pottmann, L. Shi and M. Skopenkov, Computer Aided Geometric Design, 29(1):77-79, 2012, [arxiv] |

[2] |
Surfaces that are covered by two families of circles
N. Lubbes, [arxiv] |

[3] |
Möbius automorphisms of surfaces with many circles
N. Lubbes, Canadian Journal of Mathematics, 2020, [journal], [arxiv], |

[4] |
Quadratic solutions of quadratic forms
J. Kollár, Local and global methods in algebraic geometry, volume 712 of Contemporary Mathematics, pages 211-249, 2018, [arxiv] |

[5] |
Surfaces containing two circles through each point
M. Skopenkov and R. Krasauskas, Mathematische Annalen, 2018, [journal] |

[6] |
Euclidean sums and Hamiltonian products of circles in the 3-sphere
N. Lubbes, [arxiv] |