Variational Constructions using Algebraic Curves and Surfaces

Written by Pavel Chalmovianský (external)

doc. RNDr. Pavel Chalmovianský, PhD.

Fakulta matematiky, fyziky a informatiky
Univerzita Komenského, Bratislava, Slovensko

Invited talk in Slovak
May 22, 2012 at 12:00
University of West Bohemia, UK417

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Abstract

We discuss certain constructions of variational design. They are applied to modify/approximate curves/surfaces with respect to the given criteria such as distance to the set of fixed point and/or set of fixed normals at these points. For the modeling/approximation, we use mostly algebraic curves/surfaces often represented in rational form. The applications covered by these constructions are filling holes in point clouds, approximate parameterization of plane/spatial curves, non-linear subdivision scheme. In these constructions, we consider also singular points of algebraic curves/surfaces and state several properties of such points not only due to their significant influence during the optimization process on the run of the optimization. It seems to be a difficult task to understand the structure of singularities even in low dimensional cases and low degree algebraic varieties. We discuss several notions and give visual examples for them.