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]).
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Translations of circles in Euclidean and elliptic space,
Submitted, arXiv:1302.6710, 2013