On the numerical condition of Bernstein-Bézier subdivision processes

Abstract

The linear map M that takes the Bernstein coefficients of a polynomial P ( t ) P(t) on a given interval [a, b] into those on any subinterval [ a ¯ , b ¯ ] [\bar a,\bar b] is specified by a stochastic matrix which depends only on the degree n of P ( t ) P(t) and the size and location of [ a ¯ , b ¯ ] [\bar a,\bar b] relative to [a, b]. We show that in the ‖ ∙ ‖ ∞ {\left \| \bullet \right \|_\infty } -norm, the condition number of M has the simple form κ ∞ ( M ) = [ 2 f max ( u m ¯ , v m ¯ ) ] n {\kappa _\infty }({\mathbf {M}}) = {[2f\max ({u_{\bar m}},{v_{\bar m}})]^n} , where u m ¯ = ( m ¯ − a ) / ( b − a ) {u_{\bar m}} = (\bar m - a)/(b - a) and v m ¯ = ( b − m ¯ ) / ( b − a ) {v_{\bar m}} = (b - \bar m)/(b - a) are the barycentric coordinates of the subinterval midpoint m ¯ = 1 2 ( a ¯ + b ¯ ) \bar m = \frac {1}{2}( {\bar a + \bar b} ) , and f denotes the "zoom" factor ( b − a ) / ( b ¯ − a ¯ ) (b - a)/(\bar b - \bar 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 κ ∞ ( M ) {\kappa _\infty }({\mathbf {M}}) with n also argues forcefully against the use of high-degree forms in computer-aided geometric design applications.