Toric Degenerations of Bézier Patches: Extended Abstract

In geometric modeling of curves and surfaces, the overall shape of an individual patch is intuitively governed by the placement of control points, and a patch may be finely tuned by altering the weights of the basis functions—large weights pull the patch towards the corresponding control points. The control points also have a global meaning as the patch lies within the convex hull of the control points, for any choice of weights. This convex hull is often indicated by drawing some edges between the control points. The rational bicubic tensor product patches in Figure 1 have the same weights but different control points, and the same 3 × 3 quadrilateral grid of edges drawn between the control points. Unlike the control points or their convex hulls, there is no canonical choice of these