Some Geometrical Aspects of Control Points for Toric Patches

Purchase on Springer.com

$29.95 / €24.95 / £19.95*

* Final gross prices may vary according to local VAT.

Get Access

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.