On the numerical condition of Bernstein-Bézier subdivision processes

Authors:
R. T. Farouki and C. A. Neff

Journal:
Math. Comp. **55** (1990), 637-647

MSC:
Primary 65D10; Secondary 65D15, 68T10, 68U05

DOI:
https://doi.org/10.1090/S0025-5718-1990-1035933-0

MathSciNet review:
1035933

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Abstract: The linear map **M** that takes the Bernstein coefficients of a polynomial on a given interval [*a, b*] into those on any subinterval is specified by a stochastic matrix which depends only on the degree *n* of and the size and location of relative to [*a, b*]. We show that in the -norm, the condition number of **M** has the simple form , where and are the barycentric coordinates of the subinterval midpoint , and *f* denotes the "zoom" factor 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 with *n* also argues forcefully against the use of high-degree forms in computer-aided geometric design applications.

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DOI:
https://doi.org/10.1090/S0025-5718-1990-1035933-0

Article copyright:
© Copyright 1990
American Mathematical Society