Linear precision for multi-sided toric patches

Frank Sottile
Texas A&M University
Technische Universitaet Muenchen

   In 2002, Krasauskas generalized the standard Bezier
and tensor product patches of geometric modeling to
multi-sided toric patches.   While these offer the 
promise of greater design flexibility, it is not clear
whether they possess the desirable properties of
the standard patches.   One such property is linear
precision, which is the ability to replicate a linear 
function.

   In this talk, I will discuss ongoing work with Luis
Garcia on linear precision.   Our geometric analysis
separates the shape of a patch (given by a polytope
and a system of weights) from its parametrization.
We first show that for any system of weights, there 
is a unique parametrization by algebraic functions that
has linear precision.  The existence of a rational
parametrization with linear precision becomes a geometric
statement concerning the intersection of a toric variety
with a particular linear subspace (which depends on the
system of weights).  We show that the system of weights
in the classical Bezier and tensor-product patches are
essentially the unique weights having such a rational 
parametrization for polytopes which are products of simplices.