Implicitizing rational surfaces with base points using the method of moving surfaces

The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M . The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. For example, a double base point has the effect (in this new version) of converting two moving quadrics into moving planes, eliminating one additional moving quadric, and eliminating a column of the matrix (i.e., a blending function of the moving surfaces)—thereby dropping the degree of the implicit equation by four. Furthermore, this is a unifying approach whereby tensor product surfaces, pure degree surfaces, and “corner-cut” surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k − 1. Thus, we draw moving surface blending functions from the derivative ideal I′, where I is the ideal of the parametric equations. We explain the general outline of the method and show how it works in some specific cases. The paper concludes with a discussion of the method from the point of view of commutative algebra. To Bruno Buchberger in honor of his achievements in computational algebra