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

Surfaces that contain two lines through each point form hyperboloid structures and are of interest to architects. The Kobe Port tower and Shukov Tower illustrated below are examples of hyperboloid structures.

Kobe Port Tower | Shukov Tower |

The ring torus contains four circles through a general point. The diagonal circles are called Villarceau circles. Villarceau published about these circles in 1848. The rightmost image below denotes a staircase at the Strasbourg Cathedral that has sculptors with Villarceau circles. The construction of the cathedral took place between 1176 and 1439.

ring torus | Villarceau circles | Strasbourg Cathedral |

Imagine a surface being traced out by moving a circle through space, such that the change of position and radius is almost always continuous. The radius might be infinite (a line), but is almost always nonzero (a point). A celestial surface can be generated as above in at least two ways, such that both generations have at most a finite number of common circles.

For example the animation left below shows how an ellipsoid is traced out by moving a white and a black circle through space. Note that a circle degenerates to a point two times, while tracing out the surface. Below on the right we see how a circle can degenerate to a line as well. The surface on the right contains six circles through each point! For more animations see Link 1, Link 2 and Link 3.

Ellipsoid | Blum's cyclide |

Below we see an example of a translation of a circle along a circle in three dimensional Euclidean space such that each circle is parallel. Such surfaces are the pointwise sum of circles in Euclidean 3-space.

Translational surfaces in Euclidean space have a counterpart in elliptic geometry. In elliptic geometry these translations are known as Clifford translations and are realized as the pointwise Hamiltonian product of circles in the unit-quaternions. A Clifford torus is the (left/right) Clifford translation of a great circle along a great circle in the three-sphere. The ring torus above is the stereographic projection of a Clifford torus and the Villarceau circles correspond to great circles. Below we see an example of the stereographic projection of the (left/right) Clifford translation of a great circle along a little circle.

Celestial surfaces in the three-sphere are either of degree two, four or eight [2]. Below we see all the possible shapes of celestial surfaces of degree eight that contain a great circle through a general point [1].

- Clifford translation of a great circle along a great circle
- Clifford translation of a great circle along a little circle
- Clifford translation of a little circle along a little circle

The projective three-sphere is our model for three-dimensional Moebius-, elliptic- and Euclidean- geometry. All figures are stereographic projections of celestial surfaces embedded in the three-sphere.

The Moebius transformations are the projective automorphisms of the three-sphere.

The elliptic transformations are defined as the Moebius transformations that fix the hyperplane at infinity and correspond to rotations of the three-sphere (where we implicitly identify the antipodal points). The Clifford translations are defined as elliptic transformations that are isoclinic rotations.

The Euclidean transformations in our three-sphere model (!) are defined as Moebius transformations that fix a tangent hyperplane of the three-sphere. The center of stereographic projection is the contact point of this tangent hyperplane.

See the following web sites to get some feeling for the geometry of the three-sphere:

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