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Article: Computing singular points of plane rational curves

TitleComputing singular points of plane rational curves
Authors
KeywordsΜ-Basis
Implicitization
Inversion Formula
Rational Parametric Curve
Singular Point
Issue Date2008
PublisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jsc
Citation
Journal Of Symbolic Computation, 2008, v. 43 n. 2, p. 92-117 How to Cite?
AbstractWe compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the μ-basis of the curve. It is shown that singularity factors, which are defined and uniquely determined by the elementary divisors of the implicitization matrix, contain all the information about the singular points, such as the parameter values of the singular points and their multiplicities. Based on this observation, an efficient and numerically stable algorithm for computing the singular points is devised, and inversion formulae for the singular points are derived. In particular, high order singular points can be detected and computed effectively. This approach based on singularity factors can also determine whether a rational curve has any non-ordinary singular points that contain singular points in its infinitely near neighborhood. Furthermore, a method is proposed to determine whether a singular point is ordinary or not. Finally, a conjecture in [Chionh, E.-W., Sederberg, T.W., 2001. On the minors of the implicitization bézout matrix for a rational plane curve. Computer Aided Geometric Design 18, 21-36] regarding the multiplicity of the singular points of a plane rational curve is proved. © 2007 Elsevier Ltd. All rights reserved.
Persistent Identifierhttp://hdl.handle.net/10722/152384
ISSN
2015 Impact Factor: 1.03
2015 SCImago Journal Rankings: 0.979
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorChen, Fen_US
dc.contributor.authorWang, Wen_US
dc.contributor.authorLiu, Yen_US
dc.date.accessioned2012-06-26T06:37:50Z-
dc.date.available2012-06-26T06:37:50Z-
dc.date.issued2008en_US
dc.identifier.citationJournal Of Symbolic Computation, 2008, v. 43 n. 2, p. 92-117en_US
dc.identifier.issn0747-7171en_US
dc.identifier.urihttp://hdl.handle.net/10722/152384-
dc.description.abstractWe compute the singular points of a plane rational curve, parametrically given, using the implicitization matrix derived from the μ-basis of the curve. It is shown that singularity factors, which are defined and uniquely determined by the elementary divisors of the implicitization matrix, contain all the information about the singular points, such as the parameter values of the singular points and their multiplicities. Based on this observation, an efficient and numerically stable algorithm for computing the singular points is devised, and inversion formulae for the singular points are derived. In particular, high order singular points can be detected and computed effectively. This approach based on singularity factors can also determine whether a rational curve has any non-ordinary singular points that contain singular points in its infinitely near neighborhood. Furthermore, a method is proposed to determine whether a singular point is ordinary or not. Finally, a conjecture in [Chionh, E.-W., Sederberg, T.W., 2001. On the minors of the implicitization bézout matrix for a rational plane curve. Computer Aided Geometric Design 18, 21-36] regarding the multiplicity of the singular points of a plane rational curve is proved. © 2007 Elsevier Ltd. All rights reserved.en_US
dc.languageengen_US
dc.publisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/jscen_US
dc.relation.ispartofJournal of Symbolic Computationen_US
dc.subjectΜ-Basisen_US
dc.subjectImplicitizationen_US
dc.subjectInversion Formulaen_US
dc.subjectRational Parametric Curveen_US
dc.subjectSingular Pointen_US
dc.titleComputing singular points of plane rational curvesen_US
dc.typeArticleen_US
dc.identifier.emailWang, W:wenping@cs.hku.hken_US
dc.identifier.authorityWang, W=rp00186en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1016/j.jsc.2007.10.003en_US
dc.identifier.scopuseid_2-s2.0-38849195679en_US
dc.identifier.hkuros141213-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-38849195679&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume43en_US
dc.identifier.issue2en_US
dc.identifier.spage92en_US
dc.identifier.epage117en_US
dc.identifier.isiWOS:000253703900002-
dc.publisher.placeUnited Kingdomen_US
dc.identifier.scopusauthoridChen, F=7404908180en_US
dc.identifier.scopusauthoridWang, W=35147101600en_US
dc.identifier.scopusauthoridLiu, Y=27172089200en_US

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