  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.
