Toric degenerations of Bezier Patches

   Ideally, the applications of mathematics are a
two-way-street:  not only does mathematics help to
understand problems arising in engineering, but 
that work can enrich the mathematical enterprise.
I will present a small, but textbook example of 
such interactions.  

   A question of de Boor about the meaning of certain 
control structures in geometric modeling was answered 
by Garcia-Puente, Zhu, and me by appealing to toric 
degenerations, using work of Kapranov, Sturmfels, and 
Zelevinsky on the toric variety of secondary polytopes.   
These algebraic geometry methods do not extend to 
irrational toric varieties (a natural real analytic 
generalization of toric varieties).  The work I will 
describe with Postinghel and Villamizar extends this
a natural real analytic generalization of toric 
varieties, giving a dictionary between a space of 
degenerations of an irrational toric variety and the
corresponding secondary fan of the corresponding 
point configuration.  Strengthening this dictionary
should lead to a richer theory of irrational toric varieties.