An important challenge in Geometric Modeling is to classify polytopes with rational linear precision. Equivalently, in Algebraic Statistics one is interested in classifying scaled toric varieties, also known as discrete exponential families, for which the maximum likelihood estimator can be written in closed form as a rational function of the data (rational MLE). The toric fiber product (TFP) of statistical models is an operation to iteratively construct new models with rational MLE from lower dimensional ones. In this paper we introduce TFPs to the Geometric Modeling setting to construct polytopes with rational linear precision and give explicit formulae for their blending functions. A special case of the TFP is taking the Cartesian product of two polytopes and their blending functions. The Horn matrix of a statistical model with rational MLE is a key player in both Geometric Modeling and Algebraic Statistics; it proved to be fruitful providing a characterisation of those polytopes having the more restrictive property of strict linear precision. We give an explicit description of the Horn matrix of a TFP.

翻译：几何建模中的一个重要挑战是分类具有有理线性精度的多面体。等价地，在代数统计中，人们关注的是分类缩放环面簇（也称为离散指数族），其中最大似然估计可以以数据的有理函数形式（有理MLE）闭式表达。统计模型的环面纤维积（TFP）是一种通过低维模型迭代构造具有有理MLE的新模型的操作。本文将TFP引入几何建模领域，用于构造具有有理线性精度的多面体，并给出其混合函数的显式公式。TFP的一个特例是两个多面体及其混合函数的笛卡尔积。具有有理MLE的统计模型的Horn矩阵在几何建模和代数统计中均扮演关键角色；研究表明，该矩阵能有效刻画具有更严格性质——严格线性精度的多面体。本文给出了TFP的Horn矩阵的显式描述。