Linear precision for toric surface patches

Some geometrical aspects of control points for toric patches

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 importa…

Linear precision for parametric patches

We give a precise mathematical formulation for the notions of a parametric patch and linear precision, and establish their elementary properties. We relate linear precision to the geometry of a particular linear projection, giving necessary (and quite restrictive) conditions for a patch to possess linear precision. A main focus is on linear precision for Krasauskas' toric patches, which we show is equivalent to a certain rational map on CP^d being a birational isomorphism. Lastly, we establish the connection between linear presision for toric surface patches and maximum likelihood degree for d…

Injectivity of 2D Toric Bézier Patches

Rational Bézier functions are widely used as mapping functions in surface reparameterization, finite element analysis, image warping and morphing. The injectivity (one-to-one property) of a mapping function is typically necessary for these applications. Toric Bézier patches are generalizations of classical patches (triangular, tensor product) which are defined on the convex hull of a set of integer lattice points. We give a geometric condition on the control points that we show is equivalent to the injectivity of every 2D toric Bézier patch with those control points for all possible choices of…

Toric degenerations of Bezier patches

The control polygon of a Bezier curve is well-defined and has geometric significance---there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. We propose a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary.

几何建模中的环面纤维积

Toric Fiber Products in Geometric Modeling

该研究将统计模型中的环面纤维积（TFP）引入几何建模，用于构建有理线性精度多面体并给出其混合函数、Horn矩阵的显式公式，连通了两个领域的相关研究。

Moment Maps, Strict Linear Precision, and Maximum Likelihood Degree One

We study the moment maps of a smooth projective toric variety. In particular, we characterize when the moment map coming from the quotient construction is equal to a weighted Fubini-Study moment map. This leads to an investigation into polytopes with strict linear precision, and in the process we use results from and find remarkable connections between Symplectic Geometry, Geometric Modeling, Algebraic Statistics, and Algebraic Geometry.