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.

##### MSC:

65D17 | Computer aided design (modeling of curves and surfaces) |