中文摘要：

给出一组含有3个参数的四次多项式基函数,它是三次Bernstein基函数的扩展;基于该组基定义了带形状参数的多项式曲线,称之为广义三次Bézier（GCB）曲线。GCB曲线不仅具有三次Bézier曲线的特征,而且在控制多边形保持不变的条件下,具有形状可调性和对控制多边形更好的逼近性。讨论了两条GCB曲线C2拼接的条件,并构造了C2形状可调的GCB样条曲线。图形实例表明：构造的GCB曲线为曲线曲面设计提供了有效的新方法。

英文摘要：

A new formulation for the representation and designing of curves is presented,which can be regarded as a novel generalization of cubic Bézier curves.Firstly,a class of polynomial basis functions with 3 adjustable shape parameters is present.It is a natural extension to classical Bernstein basis functions.The corresponding Bézier curves,the so-called generalized cubic Bézier（GCB）curves,are also constructed and their properties studied.It has been shown that the main advantage compared to the ordinary Bézier curves is that after inputting a set of control points and values of newly introduced 3 shape parameters,the desired curve can be flexibly chosen from a set of curves which differ either locally or globally by suitably modifying the values of the shape parameters,when the control polygon remains.The C2 GCB spline curve is constructed.The resulted curves are locally adjustable.Some examples illustrate the new curves are very valuable for the design of curves and surfaces.