Linear precision for toric patches

Frank Sottile
Texas A&M University

   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.

   I will discuss work with Luis Garcia on linear precision.   
We show that every patch has a reparametrization having
linear precision.  The reparametrization is not rational
unless the patch has a very singular geometry.  For toric 
patches, the existence of such rational reparametrizations
has an appealing mathematical reformulation in terms of 
Cremona transformations.  I will also report on work with
Ranestad and Graf von Bothmer on classifying toric 
surfaces patches with linear precision.