The concept of a \u00b5-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The \u00b5-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 \u00b5-basis to an arbitrary rational parametric surface. We show that: (1) the \u00b5-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 \u00b5-basis is in fact a basis of the moving plane module of the rational surface; and (3) the \u00b5-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 \u00b5-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gr\u00f6bner basis. Problems for further research are also discussed.<\/p>\n","protected":false},"excerpt":{"rendered":"
The concept of a \u00b5-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The \u00b5-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. 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