Implicitizing rational surfaces with base points using the method of moving surfaces
Implicitizing rational surfaces with base points using the method of moving surfaces
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DOI：
10.1090/conm/334/05980
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发表时间：
2003
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影响因子：
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通讯作者：
J. Zheng;T. Sederberg;E. Chionh;David A. Cox
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文献类型：
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作者：
J. Zheng;T. Sederberg;E. Chionh;David A. Cox
移动平面和移动二次曲面的方法可以将参数曲面的隐式方程表示为矩阵M的行列式。M的行对应于跟随参数曲面的移动平面或移动二次曲面。以前关于移动曲面方法的论文已经表明，一个简单的基点具有将一个移动二次曲面转换为一个移动平面的效果。本文提出了一种更一般的移动曲面法，它能够处理多个基点。例如，双基点具有将两个移动二次曲面转换为移动平面、消除一个额外的移动二次曲面以及消除矩阵的一列（即，移动表面的混合函数）-从而将隐式方程的阶数降低4。此外，这是一个统一的方法，张量积曲面，纯度曲面和“切角”曲面，都可以在同一框架下隐含，不需要被视为不同的情况。这种方法的中心思想是，如果一个曲面有一个重数为k的基点，那么移动曲面混合函数必须有相同的基点，但重数为k − 1。因此，我们从导数理想I′画出移动曲面混合函数，其中I是参数方程的理想。我们解释的方法的一般轮廓，并显示它是如何在一些特定的情况下工作。本文最后从交换代数的角度讨论了这种方法。以荣誉表彰布鲁诺·布赫伯格在计算代数方面的成就
The method of moving planes and moving quadrics can express the implicit equation of a parametric surface as the determinant of a matrix M . The rows of M correspond to moving planes or moving quadrics that follow the parametric surface. Previous papers on the method of moving surfaces have shown that a simple base point has the effect of converting one moving quadric to a moving plane. A much more general version of the method of moving surfaces is presented in this paper that is capable of dealing with multiple base points. For example, a double base point has the effect (in this new version) of converting two moving quadrics into moving planes, eliminating one additional moving quadric, and eliminating a column of the matrix (i.e., a blending function of the moving surfaces)—thereby dropping the degree of the implicit equation by four. Furthermore, this is a unifying approach whereby tensor product surfaces, pure degree surfaces, and “corner-cut” surfaces, can all be implicitized under the same framework and do not need to be treated as distinct cases. The central idea in this approach is that if a surface has a base point of multiplicity k, the moving surface blending functions must have the same base point, but of multiplicity k − 1. Thus, we draw moving surface blending functions from the derivative ideal I′, where I is the ideal of the parametric equations. We explain the general outline of the method and show how it works in some specific cases. The paper concludes with a discussion of the method from the point of view of commutative algebra. To Bruno Buchberger in honor of his achievements in computational algebra