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Families of circles on surfaces
, 2014
"... We classify surfaces in 3space that carry at least 2 families of real circles. Equivalently, we classify surfaces with at least 2 real circles through a generic closed point. We call such surfaces real celestials. We show that celestials are weak Del Pezzo surfaces. The Euclidean type of a surface ..."
Abstract

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We classify surfaces in 3space that carry at least 2 families of real circles. Equivalently, we classify surfaces with at least 2 real circles through a generic closed point. We call such surfaces real celestials. We show that celestials are weak Del Pezzo surfaces. The Euclidean type of a surface is a tuple (d,c) defined by the degree of the surface in 3space and the multiplicity of the Euclidean absolute in the surface. The degree of the Moebius model of a celestial of Euclidean type (d,c) is 2(dc). The Moebius model of a celestial in the 3sphere is of degree 2, 4 or 8. We show that the Euclidean type of a celestial is either (1,0) (plane), (2,1) (sphere), (2,0), (3,1), (4,2) (Darboux cyclides), (4,0), (6,2), (7,3) or (8,4). We classify celestials in 3space up to Weyl equivalence. We describe the geometry and singular loci of such celestials. As a result of our classification we obtain an alternative proof for the known fact that a real celestial carries either infinite or at
NEW EXAMPLES OF HEXAGONAL WEBS OF CIRCLES
"... Abstract. We give several new examples of hexagonal 3webs of circles in the plane and give a survey on such webs. ..."
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Abstract. We give several new examples of hexagonal 3webs of circles in the plane and give a survey on such webs.