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

The construction of Bezier 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 Bezier 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 C-0, C-1 and C-2 continuity as well as G(0), G(1) and G(2) 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 Bezier curves, but allows the shape parameters to be altered, making it a valuable method for the design of curves and surfaces. (C) 2013 Elsevier Inc. All rights reserved.