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Discrete and Computational Geometry

December 4 (Wednesday), 1300, IITP RAS, room 307

Mikhail Skopenkov

Surfaces Đontaining Several Circles Through Each Point

Motivated by potential applications in architecture, we study surfaces in 3-dimensional Euclidean space containing several circles through each point. Complete classification of such surfaces is a challenging open problem. We provide some examples and reduce the problem to an algebraic one. For the latter we provide some partial advances.

An old theorem of Darboux says that a surface containing sufficiently many circles through each point must be a so-called Darboux cyclide. Darboux cyclides are algebraic surfaces of order at most 4 and are a superset of Dupin cyclides and quadrics. They contain up to 6 circles through each point. We show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on all surfaces except spheres and planes.

Another class of surfaces with two circles through each point is obtained by a (Clifford) parallel translation of one circle along another one in space or in the 3-dimensional sphere. This class is nicely described by quaternions, and we attack general classification problem on surfaces with several circles through each point using quaternionic rational parametrizations.

Most part of the talk is elementary and is accessible for high school students. Several open problems are stated. An opportunity to see a lot of surfaces containing several circles through each point and to hold a Darboux cyclide in hands is provided.

03.12.2013 |


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