Computing ECT-B-splines recursively |
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Authors: | Günter W Mühlbach Yuehong Tang |
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Institution: | 1. Institut für Angewandte Mathematik, Universit?t Hannover, Hannover, Germany 2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, PR China
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Abstract: | ECT-spline curves for sequences of multiple knots are generated from different local ECT-systems via connection matrices.
Under appropriate assumptions there is a basis of the space of ECT-splines consisting of functions having minimal compact
supports, normalized to form a nonnegative partition of unity. The basic functions can be defined by generalized divided differences
24]. This definition reduces to the classical one in case of a Schoenberg space. Under suitable assumptions it leads to a
recursive method for computing the ECT-B-splines that reduces to the de Boor–Mansion–Cox recursion in case of ordinary polynomial
splines and to Lyche's recursion in case of Tchebycheff splines. For sequences of simple knots and connection matrices that
are nonsingular, lower triangular and totally positive the spline weights are identified as Neville–Aitken weights of certain
generalized interpolation problems. For multiple knots they are limits of Neville–Aitken weights. In many cases the spline
weights can be computed easily by recurrence. Our approach covers the case of Bézier-ECT-splines as well. They are defined
by different local ECT-systems on knot intervals of a finite partition of a compact interval a,b] connected at inner knots all of multiplicities zero by full connection matrices A
i] that are nonsingular, lower triangular and totally positive. In case of ordinary polynomials of order n they reduce to the classical Bézier polynomials. We also present a recursive algorithm of de Boor type computing ECT-spline
curves pointwise. Examples of polynomial and rational B-splines constructed from given knot sequences and given connection
matrices are added. For some of them we give explicit formulas of the spline weights, for others we display the B-splines
or the B-spline curves.
*Supported in part by INTAS 03-51-6637. |
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Keywords: | ECT-systems ECT-B-splines ECT-spline curves de-Boor algorithm |
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