title    : Circles on surfaces
author   : Niels Lubbes
keywords : families of circles, cyclides, Moebius geometry, elliptic geometry,
           weak Del Pezzo surfaces, translational surfaces

The sphere in 3-space has an infinite number of circles through any closed point. The torus has 4 circles through any closed point. Two of these circles are known as Villarceau circles ([1]). We define a 'celestial' to be a real surface with at least 2 real circles through a generic closed point. Equivalently, a celestial is a surface with at least 2 families of real circles.

In 1980 Blum [2] conjectured that a real surface has either at most 6 families of circles or an infinite number. For compact surfaces this conjecture has been proven by Takeuchi [3] in 1987 using topological methods. In 2001 Schicho [4] classified complex surfaces with at least 2 families of conics. This result together with Moebius geometry led to a classification of celestials in 3-space ([5]).

In 2012 Pottmann et al. [6] conjectured that a surface in 3-space with exactly 3 circles through a closed point is a Darboux Cyclide. We confirm this conjecture as a corollary from our classification in [5].

We recall that a translation is an isometry where every point moves with the same distance. In this talk we consider celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space. This is a natural extension of classical work by William Kingdon Clifford and Felix Klein on the Clifford torus.

Krasauskas, Pottmann and Skopenkov conjectured, that celestials in 3-space of Moebius degree 8 are Moebius equivalent to an Euclidean or Elliptic translational celestial. This conjecture is true if its Moebius model has a family of great circles ([7]). Moreover, its real singular locus consist of a great circle. As a corollary we obtain a classically flavored theorem in elliptic geometry: if we translate a line along a circle but not along a line then exactly 2 translated lines will coincide ([7]).


[1] M. Berger,
    Geometry revealed, A Jacob's Ladder to Modern Higher Geometry,
    Springer, 2010

[2] R. Blum,
    Circles on surfaces in the Euclidean 3-space,
    Geometry and differential geometry, 1979

[3] N. Takeuchi,
    A closed surface of genus one in E^3 cannot contain seven circles through each point,
    Proc. Amer. Math. Soc., 1987

[4] J. Schicho,
    The multiple conical surfaces,
    Contributions to Algebra and Geometry, 2001

[5] N. Lubbes,
    Families of circles on surfaces,
    Contributions to Algebra and Geometry, 2013

[6] H. Pottmann, L. Shi, M. Skopenkov,
    Darboux cyclides and webs from circles,
    Comput. Aided Geom. Design, 2012

[7] N. Lubbes,
    Translations of circles in Euclidean and elliptic space,
    Submitted, arXiv:1302.6710, 2013