Some Geometrical Aspects of Control Points for Toric Patches

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.2000 Mathematics Subject Classification. 65D17, 14M25.

A Geometric Approach to the Global Attractor Conjecture

“…Note that δ(i) > 0 for large i, because the denominators in (14) are strictly positive by the inductive hypothesis, and the numerators are nonnegative by construction. The numerator on the right-hand side of (14) has no dependence on i and hence is O(1). Also, using the identity v j (i), x = c(i) for x ∈ Super j , the denominator on the right-hand side of (14) satisfies…”

Injectivity conditions of rational Bézier surfaces

“…This geometric condition is equivalent to the surface with no self-intersection for arbitrary choice of positive weights. However, the result in [9] only guarantees injectivity in the interior of a patch. Sottile and Zhu [10] corrected this minor flaw, at least for 2D patches.…”

“…To study dynamical systems arising from chemical reaction networks, Craciun et al [8] presented an injectivity theorem for certain mappings. Based on this theorem, Craciun et al [9] proposed a geometric condition on control points of toric Bézier surface. This geometric condition is equivalent to the surface with no self-intersection for arbitrary choice of positive weights.…”

Simplicial diffeomorphisms

“…Wachspress extended barycentric coordinates to convex polygons [16,13]. The Wachspress coordinates satisfies properties (1)(2)(3)(4)(5) and recently Floater and Kosinka proved that property (6) also holds [5]. Note that property (7) only makes sense in a simplicial setting.…”

“…But we have proved that in an adjusted rational mapping x, if all the control points are fixed and the weights are positive, then x is injective, for any dimension [1, Section 6]. Recently, we learned in [2] that this result follows from Birch's Theorem, an important theorem in the theory of maximum likelihood estimation for multinomial models. In figure 3 we can see the result of a rational simplicial diffeomorphism.…”