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Journal of Applied Mathematics
Volume 2012, Article ID 293746, 25 pages
Research Article

Approximate Implicitization Using Linear Algebra

Applied Mathematics, SINTEF, ICT, P.O. Box 124, Blindern, 0314 Oslo, Norway

Received 12 July 2011; Revised 16 October 2011; Accepted 21 October 2011

Academic Editor: Michela Redivo-Zaglia

Copyright © 2012 Oliver J. D. Barrowclough and Tor Dokken. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore well suited to floating-point implementation in computer-aided geometric design (CAGD) systems. We unify the approaches under the names of commonly known polynomial basis functions and consider various theoretical and practical aspects of the algorithms. We offer new methods for a least squares approach to approximate implicitization using orthogonal polynomials, which tend to be faster and more numerically stable than some existing algorithms. We propose several simple propositions relating the properties of the polynomial bases to their implicit approximation properties.