The μ -basis and implicitization of a rational parametric surface
The μ -basis and implicitization of a rational parametric surface
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DOI：
10.1016/j.jsc.2005.01.003
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发表时间：
2005-06
影响因子：
0.7
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中科院分区：
文献类型：
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μ基的概念于1998年在参数化曲线的情况下被引入，并于2001年推广到有理直纹曲面的情况。μ基可用于恢复参数方程以及推导有理曲线或曲面的隐式方程。此外，它还可用于表面重参数化和奇异点计算。在本文中，我们将μ基的概念推广到任意有理参数曲面。我们证明：（1）有理曲面的μ基始终存在，其几何意义在于任何有理曲面都可以表示为三个运动平面的交集，而无需考虑外来因素； (2) μ基实际上是有理曲面动平面模的基； (3)μ基是基点为局部完全相交时有理曲面对应的动面理想的基。作为副产品，提出了一种新算法，用于从 μ 基计算有理曲面的隐式方程。实例证明新算法优于基于 Gröbner 基直接计算的传统算法。还讨论了进一步研究的问题。
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 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.