Abstract: For a class of Generalized Cubic Bézier（GCB） curves with 3 shape parameters，the relationship between its basis function and quartic Bernstein basis function is deduced.The matrix form of the GCB curve is given.The geometric construction of GCB curve is obtained and the relationship between the GCB curve and classical quartic Bézier curves is established by degree elevation.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.Some examples illustrate the new curves are very valuable for the design of curves and surfaces.

Key words: quartic Bézier curve, GCB curve, shape parameters, degree elevation, Bernstein basis function

摘要： 针对一类含有3个形状参数的广义三阶Bézier（GCB）曲线，推导出GCB曲线的基函数与四次Bernstein基函数的转换公式。利用升阶公式，建立了它与四次Bézier曲线的关系，给出了几何结构和矩阵表示形式。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。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。

关键词: 四次Bézier曲线, GCB曲线, 形状参数, 升阶公式, Bernstein基函数