Implicitization of Rational Surfaces with Base Points

本文的目的是研究一种新的求有理曲面隐式方程的方法。 有理曲面的标准隐式化技术基于结式。 但在基点的存在下，结式消失了。 因此，当基点存在时，隐式方程可以通过仔细扰动参数方程以消除基点来使用结式恢复。 但这种扰动通常会引入外部因素，然后需要精心消除。 因此，基点严重复杂的隐式使用结式。 另一方面，基点通过降低有理曲面的次数来简化其隐式方程。 初步工作已经粗略地提出了一个新的隐式化技术的基础上代数“移动曲面”，“遵循”一个合理的表面。 这种新的隐式化技术不仅不受基点的影响，而且在基点存在的情况下也进行了简化。 此外，这种新方法产生了一个更紧凑的矩阵表示的隐式方程比基于结式的方法。 目前这种新的方法只是通过几个Mathematica的例子和一些初步的定理来证明。 建议开发新方法的理论基础，然后应用该方法来解决计算机图形学，几何建模和符号计算中的各种实际问题。 消除理论中的各种其他问题也可能受到这些新技术的攻击，包括但不限于，反转有理映射，计算有理曲面的奇异轨迹，相交有理曲线和曲面，相交代数曲线和曲面。 ***

The objective of this proposal is to investigate a new technique for finding the implicit equation of a rational surface. Standard implicitization techniques for rational surfaces are based on resultants. But resultants vanish in the presence of base points. Therefore, when base points are present, the implicit equation can be recovered using resultants only by carefully perturbing the parametric equations to eliminate the base points. But such perturbations generally introduce extraneous factors, which then need to be painstakingly removed. Thus base points seriously complicate implicitization using resultants. On the other hand, base points simplify the implicit equation of a rational surface by lowering its degree. Preliminary work has roughed out a new implicitization technique based on algebraic "moving surfaces" that "follow" a rational surface. Not only is this new implicitization technique impervious to base points, it also simplifies in the presence of base points. Moreover this new method generates a more compact matrix representation for the implicit equation than methods based on resultants. Currently this new method is substantiated only by several Mathematica examples and some preliminary theorems. It is proposed to develop the theoretical foundations of the new method and then to apply the method to solve a variety of practical problems in computer graphics, geometric modeling, and symbolic computation. A wide variety of additional problems in elimination theory might also be attacked by these new techniques including, but not limited to, inverting a rational map, computing the singular locus of a rational surface, intersecting rational curves and surfaces, and intersecting algebraic curves and surfaces. ***