Determining the Regularity of Bézier Curve and Surface by Gröbner Bases Method

The determination of regularity of Bezier curves and surfaces can be converted to the problem of solutionpsilas existence of an overdetermined polynomial equations system. According to Hilbertpsilas weak Nullstellensatz, we compute the reduced Grobner basis G of the ideal generated by the tangent (normal) vector equations of Bezier curve (surface) so as to determine the regularity of curves or surfaces. If G = {1}, it means the curve (surface) is regular; or else it is irregular. Comparing with other method, the condition for determining the regularity is very simple in our methods. Meanwhile, It is avoided to implicit the parametric equations and compute the value of the large scale of determinants. Besides these, this method can compute the symbolic solution of parameters at all singular points if Bezier curve or surface is irregular. We can get the numeric solution by using real root isolation or other method for further.