参数曲线曲面实奇异点的计算

本文主要讨论了利用Grobner基理论对参数曲线（面）的奇异点进行判断和计算。如果曲线（面）存在奇异点，由定义可知它的导矢（法矢）等于0。因此，曲线（面）奇异点的判定就是方程组的求解问题。由Hilbert弱零点定理可知，若一组多项式方程无公共零点，则其生成理想约化的Grobner基为1]。在计算时，首先根据Grobner基理论判断 曲线（面）是否存在奇异点。当存在奇异点时，利用区间算法对实奇异点进行隔离和迭代。在确定奇异点的存在性时，根据曲线（曲面）的导矢（法矢）方程的Grobner基直 接进行判断，而不需要求解非线性代数方程组。若曲线曲面存在奇异点，进一步采用区间方法对奇异点进行隔离以确定曲线段或曲面片的正则性。该方法可以得到参数曲线曲面的所有实奇异点且达到任意精度。

Computation of the Singular Points of Parametric Curves and Surfaces

The paper discusses how to determine and compute the singular points of parametric curves and surfaces by using the Grobner basis theory.According to the definitions,if there exists singular points on a curve(surface),then its tangent(normal)vector is equal to 0.Therefore,we can determine the singular points' existence by solving equations.However,how can we determine whether it exists on curves or surfaces but not solving equations-According to the Hilbert Weak Nullstellensatz,if the polynomial equations fail to have a common zero point,then the reduced Grobner basis of its generating ideal is 1].We determine whether there exist singular points by this theorem.Furthermore,we isolate all real singular points by an interval algorithm when they appear on the curves or surfaces.This method can determine the existence of singular points but not solving nonlinear equations.As the interval method is introduced,we can find all real singular points with any arbitrary precision when there exist singular points on the curves or surfaces.