The mu-basis and implicitization of a rational parametric surface
The concept of a &mu;-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The &mu;-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a &mu;-basis to an arbitrary rational parametric surface. We show that: (1) the &mu;-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the &mu;-basis is in fact a basis of the moving plane module of the rational surface; and (3) the &mu;-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the &mu;-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Grobner basis. Problems for further research are also discussed. &COPY; 2005 Elsevier Ltd. All rights reserved.
mu-basis implicitization rational surface moving plane base point syzygy module