Generalized Cubic Bézier Curves and its Application

A new formulation for the representation and designing of curves is presented,which can be regarded as a novel generalization of cubic Bezier 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 Bezier curves,the so-called generalized cubic Bezier(GCB)curves,are also constructed and their properties studied.It has been shown that the main advantage compared to the ordinary Bezier 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.