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) |
Bohemian and Cliffordian celestial surfaces
Bohemian celestial surfaces are the pointwise vector sum of two circles and/or lines in Euclidean 3-space. The Cliffordian celestial surfaces are their counterpart in elliptic geometry, namely the pointwise Hamiltonian product of circles in the unit 3-sphere S3, where we identified S3 with the unit-quaternions. For example, Bohemian domes and Clifford tori are Cliffordian and Bohemian celestial surfaces, respectively.
Mikhail Skopenkov and Rimvydas Krasauskas showed in [5] that a celestial surface in 3-space is up to Möbius equivalence either
- Cliffordian, Bohemian or a Darboux cyclide.
We showed in [6, Theorem 1], that a Bohemian Darboux cyclide is either a
- plane, circular cylinder or elliptic cylinder.
Moreover, a Cliffordian Darboux cyclide is either a
Below we rendered some examples of Bohemian celestial surfaces and stereographic projections of Cliffordian celestial surfaces.
See the following websites to get some feeling for the geometry of the 3-sphere:
- Möbius 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 pencils of circles,
N. Lubbes, Mathematische Zeitschrift, 2021, [journal], [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] | Translational and great Darboux cyclides, N. Lubbes, [arxiv] |