The number of regular control surfaces of toric patch

Through a rational map, a toric patch is defined associated to a lattice polygon, which is the convex of a given finite integer lattice points set A. The classical rational Bézier curves, rational triangular and tensor-product patches are special cases of toric patches. One of the geometric meanings of toric patch is that the limiting of the patch is its regular control surface, when all weights tend to infinity. In this paper, we study the number of regular decompositions of A, and the relationship between regular decompositions and the corresponding secondary polytope. What is more, we indicate that the number of regular control surfaces of toric patch associated with A is equal to the number of regular decompositions of A.