A Direct Approach to Computing theμ-basis of Planar Rational Curves

Zheng, Jianmin; Sederberg, Thomas W.

doi:10.1006/jsco.2001.0437

Cited by 29 publications

(10 citation statements)

References 12 publications

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“…To build such a matrix it is necessary to first compute a µ-basis of the parameterization of the curve. There exist efficient algorithms to compute µ-basis (see [24,21]), but they all require the use of exact linear algebra routines. Therefore, in order to make matrix representations accessible to any programming environment having linear algebra routines (but not necessarily exact), we provide a new family of matrix representations that does not require symbolic computations to be built.…”

Section: Complement: Matrix Representations Without µ-Basesmentioning

confidence: 99%

Matrix-based implicit representations of rational algebraic curves and applications

Busé

2010

Computer Aided Geometric Design

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Given a parameterization of an algebraic rational curve in a projective space of arbitrary dimension, we introduce and study a new implicit representation of this curve which consists in the locus where the rank of a single matrix drops. Then, we illustrate the advantages of this representation by addressing several important problems of Computer Aided Geometric Design: The point-on-curve and inversion problems, the computation of singularities and the calculation of the intersection between two rational curves.

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Section: Complement: Matrix Representations Without µ-Basesmentioning

confidence: 99%

Matrix-based implicit representations of rational algebraic curves and applications

Busé

2010

Computer Aided Geometric Design

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“…2 The variety of possible applications motivates the development of algorithms for computing µ-bases. Although a proof of the existence of a µ-basis for arbitrary n appeared already in [7], the algorithms were first developed for the n = 3 case only [7,14,5]. The first algorithm for arbitrary n appeared in [12], as a generalization of [5].…”

Section: Introductionmentioning

confidence: 99%

“…Relation to the previous algorithms: Cox, Sederberg and Chen [7] implicitly suggested an algorithm for the n = 3 case. Later, it was explicitly described in the Introduction of [14]. The algorithm relies on the fact that, in the n = 3 case, there are only two elements in a µ-basis, and their degrees (denoted as µ 1 and µ 2 ) can be determined prior to computing the basis (see Corollary 2 on p. 811 of [7] and p. 621 of [14]).…”

Section: Introductionmentioning

confidence: 99%

Algorithm for computing μ-bases of univariate polynomials

Hong

Hough

Kogan

2017

Journal of Symbolic Computation

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We present a new algorithm for computing a µ-basis of the syzygy module of n polynomials in one variable over an arbitrary field K. The algorithm is conceptually different from the previously-developed algorithms by Cox, Sederberg, Chen, Zheng, and Wang for n = 3, and by Song and Goldman for an arbitrary n. The algorithm involves computing a "partial" reduced row-echelon form of a (2d+ 1)×n(d+ 1) matrix over K, where d is the maximum degree of the input polynomials. The proof of the algorithm is based on standard linear algebra and is completely self-contained. The proof includes a proof of the existence of the µ-basis and as a consequence provides an alternative proof of the freeness of the syzygy module. The theoretical (worst case asymptotic) computational complexity of the algorithm is O(d 2 n + d 3 + n 2 ). We have implemented this algorithm (HHK) and the one developed by Song and Goldman (SG). Experiments on random inputs indicate that SG is faster than HHK when d is sufficiently large for a fixed n, and that HHK is faster than SG when n is sufficiently large for a fixed d.

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“…This method needs O(n 3 ) arithmetic operations, where n is the degree of the curve, and it is a trial-and-error approach. The second method was developed by Zheng and Sederberg [18], and it is similar to the Buchberger's algorithm for computing the Gröbner basis of a module. The computational cost of the method is about 81 4 n 2 + O(n) multiplications in generic case.…”

Section: Introductionmentioning

confidence: 99%

Computing μ-bases of rational curves and surfaces using polynomial matrix factorization

Deng

Chen

Shen

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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The µ-bases of rational curves/surfaces are newly developed tools which play an important role in connecting parametric forms and implicit forms of the rational curves/surfaces. They provide efficient algorithms to implicitize rational curves/surfaces as well as algorithms to compute singular points of rational curves and to reparametrize rational ruled surfaces. In this paper, we present an efficient algorithm to compute the µ-basis of a rational curve/surface by using polynomial matrix factorization followed by a technique similar to Gaussian elimination. The algorithm is shown superior than previous algorithms to compute the µ-basis of a rational curve, and it is the only known algorithm that can rigorously compute the µ-basis of a general rational surface. We present some examples to illustrate the algorithm.

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A Direct Approach to Computing theμ-basis of Planar Rational Curves

Cited by 29 publications

References 12 publications

Matrix-based implicit representations of rational algebraic curves and applications

Matrix-based implicit representations of rational algebraic curves and applications

Algorithm for computing μ-bases of univariate polynomials

Computing μ-bases of rational curves and surfaces using polynomial matrix factorization

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