On the numerical condition of bernstein-bÉzier subdivision processes

Abstract

The linear map M that takes the Bernstein coefficients of a polynomial P(t) on a given interval [a, b] into those on any subinterval [a, b] is specified by a stochastic matrix which depends only on the degree n of P(t) and the size and location of [ä, b] relative to [a, b]. We show that in the • H^-norm, the condition number of M has the simple form «^(M) = [2/max(w-, vm)]n, where um = (m - a)/(b - a) and vw=(b- m)/(b - a) are the barycentric coordinates of the subinterval midpoint m = j(3 + è), and denotes the "zoom" factor (b-a)/(b-a) of the subdivision map. This suggests a practical rule-of-thumb in assessing how far Bézier curves and surfaces may be subdivided without exceeding prescribed (worst-case) bounds on the typical errors in their control points. The exponential growth of ^(M) with n also argues forcefully against the use of high-degree forms in computer-aided geometric design applications. © 1990 American Mathematical Society.