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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 [a, T] relative to [a, b]. We show that in the 11 * 110j-norm, the condition number of M has the simple form K. (M) = [2f max(u7, VO)], where u5 = (m a)/(b a) and v, = (b m)/(b a) are the barycentric coordinates of the subinterval midpoint m = 2 (a + b) , and f denotes the "zoom" factor (b a)/(b a) of the subdivision map. This suggests a practical rule-of-thumb in assessing how far Bezier curves and surfaces may be subdivided without exceeding prescribed (worst-case) bounds on the typical errors in their control points. The exponential growth of Koo (M) with n also argues forcefully against the use of high-degree forms in computer-aided geometric design applications.

On the numerical condition of Bernstein-Bézier subdivision processes