Abstract: In order to construct Bézier curves with shape control parameters，two classes of polynomial basis functions of 3th and 4th degree are presented in this paper.They are both extensions of cubic Bernstein basis functions.Properties of the proposed basis functions are analyzed and the corresponding polynomial curves with shape control parameters α，γ and α，β，γ are constructed accordingly.The constructed curves not only inherit the outstanding properties of the cubic Bézier curve，but also are adjustable in shape and fit close to the control polygon.In addition，the continuity condition and some applications in curve and surface design about the new curves are discussed，and two definition of Extension-surface are also presented.Some examples illustrate the new curves are very valuable for the design of curves and surfaces.

Key words: Bézier curve, curve design, shape parameter, extension, continuity

摘要： 给出了两组分别含有2个和3个形状控制参数的三次和四次多项式基函数，它们都是三次Bernstein基函数的扩展；分析了这两组基函数的性质，基于此两组基定义了两种分别带形状参数α，γ和α，β，γ的多项式曲线，它们都以三次Bézier曲线为特殊情形。两种新曲线不仅具有三次Bézier曲线的特性，而且具有灵活的形状可调性和更好的逼近性。最后讨论了两种扩展曲线的拼接条件及它们在曲线曲面造型中的应用，并给出了两个扩展曲面的定义。实例表明，定义的两种新扩展曲线为曲线/曲面的设计提供了两种有效的新方法。

关键词: Bézier曲线, 曲线设计, 形状参数, 扩展, 拼接