Some geometrical aspects of control points for toric patches

Details Some geometrical aspects of control points for toric patches Some geometrical aspects of control points for toric patches

2008-12-06

arxiv.org/abs/0812.1275v2

Mathematics

Algebraic Geometry

24 pages, many color figures

Scientific paper

We use ideas from algebraic geometry and dynamical systems to explain some ways that control points influence the shape of a B\'ezier 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 B\'ezier and tensor product patches as important special cases.

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