An extension of Bézier curve

In CAGD,in order to adjust the shape of curves or change the site of curves easily,it is necessary to get a method of generating piecewise polynomial curves with a shape parameter. In this paper, a class of polynomial blending functions of n+1 degrees are presented,which are the extension of the Bernstein basis functions of n degrees. Based on the presented functions,the polynomial curves with a shape parameter are constructed. The properties of the basis functions, such as the weighting property, positivity, symmetry and endpoints' properties,are similar with those of the n-degree Bernstein basis functions,and the generated curves are similar with the n-degree Bezier curves. By changing the value of the shape parameter,the approaching degree of the curves to their control polygons can be adjusted and the n-degree Bezier curves can be manipulated from both sides. The presented method is very useful for the curve design.