带形状参数的二次三角多项式Bézier曲线形状分析

Shape analysis of quadratic trigonometric polynomial Bézier curves with a shape parameter

对一类二次三角多项式Bézier曲线的形状及其控制多边形之间的关系进行了研究.根据控制多边形边之间的相对位置关系,先通过计算推理得到有关空间二次三角多项式Bézier曲线奇、拐点的一个结论;再利用包络理论和拓扑映射的方法,分别得到平面二次三角多项式Bézier曲线上含有尖点、拐点、重结点和曲线为全局凸、局部凸的充分必要条件,并给出了曲线具有尖点、重结点和拐点的数值例子;最后,讨论了形状参数对形状分区的影响.

The relationship of the shape features between a class of quadratic trigonometric polynomial Bezier curves and their control polygons are investigated. According to the relationship between control polygons and its edge vectors, a result on the singularity and inflection points of the space quadratic trigonometric polynomial Bezier curves is obtained by calculation and deduction, and then the necessary and sufficient conditions for cusps, loops and inflection points and locally or globally convex of the planar quadratic trigonometric polynomial Bezier curves are obtained respectively by using the method based on the theory of envelop and topological mapping, some numerical examples of the curves with cusp, loop and inflection point are also given. Finely the influences of shape parameter to the shape distribution are discussed.