A novel extension to the polynomial basis functions describing Bezier curves and surfaces of degree n with multiple shape parameters

Abstract

The construction of Bézier curves using shape control parameters is one of the most popular areas of research in computer aided geometric design (CAGD). A class of new polynomial basis functions with n−1 local shape control parameters is presented here to allow the construction of Bézier curves with n local shape control parameters, which is an extension to the classical Bernstein basis functions of degree n. The properties of the proposed basis functions and the corresponding piecewise polynomial curve with n−1 local shape control parameters are analyzed. This analysis shows that the new class of polynomial functions meets the conditions required for both C0, C1 and C2 continuity as well as G0, G1 and G2 continuity. Some curve design applications are then discussed and an extended application for surface design is also presented. The modeling examples illustrate that the new extension provides not only a better approximation and mathematical description of Bézier curves, but allows the shape parameters to be altered, making it a valuable method for the design of curves and surfaces.