The objective of this proposal is to investigate a new technique for finding the implicit equation of a rational surface. Standard implicitization techniques for rational surfaces are based on resultants. But resultants vanish in the presence of base points. Therefore, when base points are present, the implicit equation can be recovered using resultants only by carefully perturbing the parametric equations to eliminate the base points. But such perturbations generally introduce extraneous factors, which then need to be painstakingly removed. Thus base points seriously complicate implicitization using resultants. On the other hand, base points simplify the implicit equation of a rational surface by lowering its degree. Preliminary work has roughed out a new implicitization technique based on algebraic "moving surfaces" that "follow" a rational surface. Not only is this new implicitization technique impervious to base points, it also simplifies in the presence of base points. Moreover this new method generates a more compact matrix representation for the implicit equation than methods based on resultants. Currently this new method is substantiated only by several Mathematica examples and some preliminary theorems. It is proposed to develop the theoretical foundations of the new method and then to apply the method to solve a variety of practical problems in computer graphics, geometric modeling, and symbolic computation. A wide variety of additional problems in elimination theory might also be attacked by these new techniques including, but not limited to, inverting a rational map, computing the singular locus of a rational surface, intersecting rational curves and surfaces, and intersecting algebraic curves and surfaces. ***
