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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The apolar bilinear form in geometric modeling


Author: Gert Vegter
Journal: Math. Comp. 69 (2000), 691-720
MSC (1991): Primary 41A15, 65D17; Secondary 65D07, 41A63
Published electronically: April 28, 1999
MathSciNet review: 1654006
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Abstract: Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives.


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Additional Information

Gert Vegter
Affiliation: Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands
Email: gert@cs.rug.nl; http://www.cs.rug.nl/~gert

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01144-8
PII: S 0025-5718(99)01144-8
Keywords: Apolar bilinear form, polarization, homogeneous polynomials, lineal polynomials, dual basis, Euler's identity, Marsden's identity, Bernstein-B{\'e}zier patches, B-patches, de Casteljau, de Boor, recurrence relations, algorithm, basis conversion
Received by editor(s): April 30, 1998
Published electronically: April 28, 1999
Article copyright: © Copyright 2000 American Mathematical Society