一类三次曲线的形状分析

摘要

This paper considers the shape features of the class of cubic curves based on the theory of envelop and topological mapping. Necessary and sufficient conditions are derived for this curve having one or two inflection points, a loop or a cusp, or be locally or globally convex. Those conditions are completely decided by control polygon and the shape parameter. Furthermore, it discusses the influences of shape parameter on the shape distribution diagram and the ability for adjusting the shape of the curve.%基于包络理论与拓扑映射的方法,对一类带有形状参数的三次曲线进行了形状分析,得出了曲线上含有奇点、拐点和曲线为局部凸或全局凸的充分必要条件,这些条件完全由控制多边形和形状参数所决定.进一步讨论了形状参数对形状分布图的影响及其对曲线形状的调节能力.