带两个形状参数的Bézier曲线

Bézier Curves with Two Shape Parameters

首先将二次Bézier曲线的基函数进行扩展,定义了带两个形状参数的三次多项式基函数,它以二次Bernstein基函数和三次λ-B基为特例。再利用德卡斯特里奥算法进行递推,得到了一般n次Bézier曲线基函数的扩展,它由n+1个带有形状参数的n+1次多项式组成。基于这组基函数定义了带有两个形状参数的多项式曲线,它以一般n次Bézier曲线和n+1次λ-Bézier曲线为特例。分析了这组基函数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。由于带有两个形状参数,这种曲线具有更加灵活的形状控制能力。

First a class of cubic polynomial functions with two parameters is presented as an extension of the quadratic Bernstein and cubic ,λ-Bernstein basis functions. Then using de Casteljau arithmetic, the generalized basis functions of degree n＋l with two shape parameters α and β is got. Based on the new basis function, a method of generating polynomial curves with two shape parameters is given. The new curves imply generalized Bézier curves and ,λ-Bézier curves. Geometric meaning of the shape parameters is given.