Some geometrical aspects of control points for toric patches. (English) Zbl 1274.65033
Dæhlen, Morten (ed.) et al., Mathematical methods for curves and surfaces. 7th international conference, MMCS 2008, Tønsberg, Norway, June 26–July 1, 2008. Revised selected papers. Berlin: Springer (ISBN 978-3-642-11619-3/pbk). Lecture Notes in Computer Science 5862, 111-135 (2010).
Summary: 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.
|65D17||Computer aided design (modeling of curves and surfaces)|