Extension of a quadratic Bézier curve

A class of polynomial functions of degree 3 with parameter λ i is presented in this paper. It is an extension of the quadratic Bezier curves basis functions. Based on the functions, a method of generating piecewise polynomial curves with a shape parameter is given. The properties and continuity conditions of the curves and its basis functions are discussed. The basis functions have weighting property and they have positivity when the parameter λ i is between -2 and 1. The curves properties are similar with the degree 2 Bezier curves, such as endpoints' properties, symmetry, convex hull, and geometric invariability. By changing value of the shape parameter λ i , the approaching degree between the i th curve and the control polygon can be adjusted. The shape parameter λ i is local, then the method is very useful for curve design.