关键词: 曲线曲面 奇异点 Grobner基 理想 区间算法

Abstract:

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 th
  curves or surfaces but not solving equations？ Ac ccording to the Hilbert Weak Nullstellensatz, if the polynomial equations fall to have a common zero point, then the reduced Grobner basis of its genera ting ideal is [1]. We determine whether there exist singular points by this theorem. Furthermore, we isolate all real singular points by an interval alg  orithm when they appear on the curves or surfaces. This method can determine the existence of singular points but not solving nonlinear equations. As th e interval method is introduced, we can find all real singular points with any arbitrary precision when there exist singular points on the curves or sur  faces.

Key words: parameter curve and surface, singular point, Grobner basis, ideal, interval algorithm