Details:
Title  The basis and implicitization of a rational parametric surface  Author(s)  Falai Chen, David A. Cox, Lu Yang  Type  Article in Journal  Abstract  The 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 byproduct, 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.  Keywords  μ basis, Moving plane, Syzygy module, Rational surface, Implicitization, Base point  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717105000398 
Language  English  Journal  Journal of Symbolic Computation  Volume  39  Number  6  Pages  689  706  Year  2005  Edition  0  Translation 
No  Refereed 
No 
