A unified approach for degree reduction of polynomials in the Bernstein basis Part I: Real polynomials

Abstract

The polynomials determined in the Bernstein (Bézier) basis enjoy considerable popularity and the algorithms for reducing their degree are of practical importance in computer aided design applications. On the other hand, the conversion between the Bernstein and the power basis is ill conditioned, thus only the degree reduction algorithms which may be carried out without using this conversion are of practical value. Our unified approach enables us to describe all the algorithms of this kind known in the literature, to construct a number of new ones, which are better conditioned and cheaper than some of the currently used ones, and to study the errors of all of them in a simple homogeneous way. All these algorithms can be applied componentwise to a vector-valued polynomial representing a Bézier curve. Consider the values of the derivatives, whose orders vary successively from 1 to a given number ι or κ at the start and end point, respectively, of this curve. The current algorithms allow us to preserve these points and values for ι equal to κ, the new ones do that also without the latter constraint.