Implicitizing Bi-Cubic Toric Surfaces by Dixon? - Resultant Quotients

Toric surface patches have two significant geometric properties: they are multi-sided and they are generalizations of both the triangular and rectangular Bezier surface patches. They also have a very nice algebraic property: their implicit equations are closely related to the Dixon determinant. In particular, for bi-cubic toric patches without base points, their implicit equation can always be obtained very conveniently using the recently discovered Dixon quotients. In this paper, we explain the relevance of monomial corner cutting to toric patches, and how this leads to their efficient implicitization by the Dixon quotient. Many examples are given to illustrate the simplicity and power of this approach.