| 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 {dollar}-{dollar} 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.; The goal of this thesis is to develop new, computationally efficient, resolvent techniques and to apply them to solve problems in Computer Aided Geometric Design (CAGD). Since these resolvent techniques are designed to be practical, they can be applied to any problem where people need to solve systems of polynomial equations.; In Part I we develop three new resolvent techniques by generalizing the Bezout, Cayley, and Sylvester resultant methods for two univariate polynomials of degree two. All three methods are more efficient than classical resolvent techniques, but the Sylvester approach is best because it generates the fewest resolvent expressions. Using the Bezout, Cayley, and Sylvester methods, we can also construct multivariate resolvents by eliminating one variable at a time.; In Part II we apply our new resolvents to solve problems in CAGD, CAGD is concerned with the design, representation, and analysis of curves, surfaces, and solids. A fundamental operation in CAGD is finding intersections of curves and surfaces, so we focus our attention on this problem.; One way to solve the intersection problem for rational curves and surfaces is to solve the implicitization and inversion problems. Implicitization means finding a set of implicit hypersurfaces whose intersection is a given parametric curve or surface: inversion means finding the point in the parameter space of the curve or surface corresponding to the ({dollar}xsb1,...,xsb{lcub}r{rcub}){dollar} coordinates of a point on the curve or surface.; By applying the Sylvester resolvent, we find that generally {dollar}r - 1{dollar} hypersurfaces of relatively low degree suffice to implicitize any r-dimensional, faithfully parametrized, polynomial curve (rational curves require some additional inequalities), and the inversion equation is also a rational expression of relatively low degree. During implicitization, we can also detect whether or not a curve is unfaithfully parametrized. For unfaithfully parametrized curves, we can obtain the composite factor, compute a faithful parametrization, and recover the implicit surfaces on which the faithfully parametrized curve lies with little additional computation. Furthermore, we can find the self intersection points of any faithfully parametrized, rational curve by using our inversion equation. |
