Emanuel Huhnen-Venedey
Member of the
Geometry Group
@ Institute of Mathematics
@ TU-Berlin Institut für Mathematik, MA 8-3, Technische Universität Berlin, Strasse des 17. Juni 136, 10623 Berlin, Germany Mail: huhnen[at]math.tu-berlin.de Phone: +49-30-314-79253 Fax: +49-30-314-79282 Room: MA 875 Office hours: on appointment Teaching Diploma thesis "Curvature line parametrized surfaces and orthogonal coordinate systems. Discretization with Dupin cyclides." (Advisor: Alexander Bobenko) The discretization is based upon so called "principal contact element nets" which were introduced by Alexander Bobenko and Yuri Suris in "On organizing principles of Discrete Differential Geometry. Geometry of spheres". Contact elements are pencils of (oriented) spheres which all touch in one point. It is the defining property of principal contact element nets that neighbouring contact elements share a common sphere. These common spheres are so to say "discrete principal curvature spheres", as the notion of principal contact element nets arises as a straight forward discretization of the Lie geometric description of curvature lines. Hence the 2-dimensional case of such nets can be seen as discretization of curvature line parametrized surfaces in Lie geometry. Additionally, elementary quadrilaterals of contact element nets carry a structure which allows pieces of Dupin cyclides to be fitted: If one starts with a principal contact element net, for each elementary quadrilateral there is a 1-parameter family of "cyclidic patches" which respect some boundary conditions determined by the contact element net. Thus one has a 1-parameter freedom for the choice of a first "cyclidic patch" associated to an arbitrary elementary quadrilateral of the given net, and it turns out that this is the starting point for a unique and consistent construction of a C^1 surface composed of such patches. The case of orthogonal coordinate systems is broken down to the 2-dimensional case by discretizing the 2-dimensional coordinate surfaces. This is done in a way such that they satisfy an orthogonality condition at vertices, namely that the boundary arcs of patches meeting in a point intersect orthogonaly. Talk at the conference "Geometry and Integrability 2008". Links
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