Bernstein-Bézier Methods for the Computer-Aided Design of Free-Form Curves and Surfaces

The m th degree Bernstein polynomial approximation to a function ƒ defined over 0, 1 is Σ m μ =0 ƒ( μ / m ) φ μ ( s ), where the weights φ μ ( s ) are binomial density functions. The Bernstein approximations inherit many of the global characteristics of ƒ, like monotonicity and convexity, and they always are at least as “smooth” as ƒ, where “smooth” refers to the number of undulations, the total variation, and the differentiability class of ƒ. Historically, their relatively slow convergence in the L ∞ -norm has tended to discourage their use in practical applications. However, in a large class of problems the smoothness of an approximating function is of greater importance than closeness of fit. This is especially true in connection with problems of computer-aided geometric design of curves and surfaces where aesthetic criteria and the intrinsic properties of shape are major considerations. For this latter class of problems, P. Bézier of Renault has successfully exploited the properties of parametric Bernstein polynomials. The purpose of this paper is to analyze the Bézier techniques and to explore various extensions and generalizations. In a sequel, the authors consider the extension of the results contained herein to free-form curve and surface design using polynomial splines . These B-spline methods have several advantages over the techniques described in the present paper.