Two Surfaces Suffice

Classical elimination theory is the study of conditions that guarantee the existence of solutions to systems of polynomial equations. The main goal is to construct a set of polynomial expressions in the coefficients of the original polynomial equations, which are satisfied exactly when the original polynomials have a common root. For n polynomials in n−1 unknowns, there is a single unique condition called the resultant. More generally, for an arbitrary number of equations and unknowns many conditions may be required; these conditions are called resolvents. Unfortunately, classical resolvents are not practical because they yield far too many conditions which are very expensive to compute.In this paper, we develop a new resolvent which is computationally efficient, and we apply it to solve the space curve/curve intersection problem in Computer Aided Geometric Design (CAGD).One way to solve the curve/curve intersection problem is to solve the implicitization and inversion problems. By applying our new resolvent, we find that generally two surfaces of relatively low degree suffice to implicitize any 3D, faithfully parametrized, polynomial curve, and the inversion equation is also a rational expression of relatively low degree. Thus we provide a very efficient curve/curve intersection algorithm for low degree, 3D, polynomial curves.