Some geometrical aspects of control points for toric patches

Authors:
Gheorghe Craciun;Luis David García-Puente;Frank Sottile
Affiliations:
Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin, Madison, WI;Department of Mathematics and Statistics, Sam Houston State University, Huntsville, TX;Department of Mathematics, Texas ASM University, College Station, TX
Venue:
MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces
Year:
2008

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Abstract

We use ideas from algebraic geometry and dynamical systems to explain some ways that control points influence the shape of a Bézier curve or patch. In particular, we establish a generalization of Birch’s Theorem and use it to deduce sufficient conditions on the control points for a patch to be injective. We also explain a way that the control points influence the shape via degenerations to regular control polytopes. The natural objects of this investigation are irrational patches, which are a generalization of Krasauskas’s toric patches, and include Bézier and tensor product patches as important special cases.