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TitleThe -basis and implicitization of a rational parametric surface
Author(s) Falai Chen, David A. Cox, Lu Yang
TypeArticle in Journal
AbstractThe concept of a μ -basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The μ -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 μ -basis to an arbitrary rational parametric surface. We show that: (1) the μ -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 μ -basis is in fact a basis of the moving plane module of the rational surface; and (3) the μ -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 μ -basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gröbner basis. Problems for further research are also discussed.
Keywordsμ -basis, Moving plane, Syzygy module, Rational surface, Implicitization, Base point
URL http://www.sciencedirect.com/science/article/pii/S0747717105000398
JournalJournal of Symbolic Computation
Pages689 - 706
Translation No
Refereed No