关于平面四次Bézier曲线的拐点与奇点

在计算几何中,已给出了三次Bezier曲线的保凸性的充要条件,并进行了几何解释。本文则是导出形式简洁的拐点和奇点方程并对四次Bezier曲线的拐点和奇点的分布进行讨论。按Bezier曲线的拐点个数进行分类,还得到了四次Bezier曲线有奇点的充分必要条件,并给出几个数值实例,实例说明,不但非凸的单纯特征多角形可以有凸的Bezier曲线段,而且非单纯特征多角形也可以有凸的Bezier曲线段。四次Bezier曲线的奇点和拐点是可以共存的。

ON DISTRIBUTION OF INFLECTION POINTS AND SINGULAR POINTS OF THE PLANAR QUARTIC B(?)ZIER CURVE

In this paper, we develop a simple equation for inflection points and singular pointsof the planar quartic Bezier curve and discuss their distribution. We suggest a clas-sification for the planar quartic Bezier curve by the number of inflection points andgive a sufficient and necessary condition for the existence of singular points (i.e. knotsand cusps). With the numerical examples we show that the convex Bezier curve canbe produced not only from the nonconvex simple characteristic polygon but also fromthe non-simple characteristic polygon and that singular points (knots) and inflectionpoints may coexist for the planar quartic Bezier curve, although this case will neveroccur for the planar cubie Bezier curve.