带形状参数的Bézier曲线

Bézier curves with shape parameters

文章首先将二次Bernstein基函数进行扩展,定义了带2个形状参数的四次多项式基函数,它以二次Bernstein基和三次-λB基为特例;再利用de Casteljau算法进行递推,得到了一般n次Bernstein基函数的扩展,它由n+1个带形状参数的n+2次多项式组成;基于这组基函数定义了带2个形状参数的多项式曲线,它以一般n次Bézier曲线和n+1次-λBézier曲线为特例;分析了这组基以及由其定义的曲线的性质,给出了形状参数的几何意义和曲线的几何作图法。

A class of polynomial functions of 4th-degree with two parameters is presented,which is the extension of quadratic Bernstein and cubic λ-B basis functions. Then using the de Casteljau arithmetic,the generalized basis function of degree n＋2 with two shape parameters is obtained. Based on this, a new type of curve with two shape parameters is defined. The new curve contains generalized Bezier curves and λ-Beizier curves. The properties of the new basis and the new curve are analyzed： Meanwhile, the geometry meaning of the shape parameters and the geometric drawing method of the new curve are given.