Bézier曲线的新扩展

New Extension of Bézier Curves

给出2组含有2个形状控制参数α,β的四次、五次多项式基函数,其分别是三次、四次Bernstein基函数的扩展。分析2组基的性质,定义带α,β的2类多项式曲线:三次E-Bézier曲线和四次E-Bézier曲线,其具有三次或四次Bézier曲线的特性、形状可调性和更好的逼近性。当α=β=0时,2类曲线分别退化为三次、四次Bézier曲线。给出2个扩展曲面的定义。实例表明,定义的曲线为曲线/曲面的设计提供了一种有效的方法。

Two classes of polynomial basis functions of the 4th and the 5th degree with two shape control parameters α,β are presented.They are extensions of cubic and quartic Bernstein basis functions.Properties of these two bases are analyzed and the corresponding polynomial curves called cubic E-Bézier curve and quartic E-Bézier curve with α,β are defined.These curves inherit the outstanding properties of cubic and quartic Bézier curve,and are better adjustable in shape and fit closer to the control polygon.These curves converge to cubic and quartic Bézier curve when α=β=0.Two definition of extension-surface are presented.Some examples illustrate the variation curve offers a sort of new effective means for design of curve/surface.