Implicitization of Rational Surfaces with Base Points

本文的目的是研究一种求有理曲面隐式方程的新方法。有理曲面的标准隐化技术是基于结果的。但在基点存在的情况下，结果会消失。因此，当基点存在时，只能通过仔细扰动参数方程来消除基点，从而使用结果恢复隐式方程。但这种扰动通常会引入外来因素，然后需要煞费苦心地去除这些因素。因此，基点严重复杂化了使用结果的隐式化。另一方面，基点通过降低有理曲面的次来简化其隐式方程。初步工作已经粗略地提出了一种新的隐式化技术，该技术基于“遵循”有理面的代数“移动面”。这种新的隐式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。与基于结果的方法相比，该方法生成了更为紧凑的隐式方程矩阵表示。目前，这种新方法仅通过几个数学实例和一些初步定理得到证实。提出了发展新方法的理论基础，然后将该方法应用于解决计算机图形学、几何建模和符号计算中的各种实际问题。这些新技术还可以解决消去理论中各种各样的附加问题，包括但不限于有理映射的反演、有理曲面的奇异轨迹计算、有理曲线与曲面的相交以及代数曲线与曲面的相交。* * *

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. ***