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Article: Computing real inflection points of cubic algebraic curves

Title Computing real inflection points of cubic algebraic curves Chen, FWang, W Algebraic curveHessian curveInflection pointInvariantSingular point 2003 Elsevier BV. The Journal's web site is located at http://www.elsevier.com/locate/cagd Computer Aided Geometric Design, 2003, v. 20 n. 2, p. 101-117 How to Cite? Shape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature. A real inflection point is also required for transforming projectively a planar cubic algebraic curve to the normal form, in order to facilitate further analysis of the curve. However, the naive method for computing the inflection points of a planar cubic algebraic curve f = 0 by directly intersecting f = 0 and its Hessian curve H(f) = 0 requires solving a degree nine univariate polynomial equation, and thus is relatively inefficient. In this paper we present an algorithm for computing the real inflection points of a real planar cubic algebraic curve. The algorithm follows Hilbert's solution for computing the inflection points of a cubic algebraic curve in the complex projective plane. Hilbert's solution is based on invariant theory and requires solving only a quartic polynomial equation and several cubic polynomial equations. Through a detailed study with emphasis on the distinction between real and imaginary inflection points, we adapt Hilbert's solution to efficiently compute only the real inflection points of a cubic algebraic curve f = 0, without exhaustive but unnecessary search and root testing. To compute the real inflection points of f = 0, only two cubic polynomial equations need to be solved in our algorithm and it is unnecessary to solve numerically the quartic equation prescribed in Hilbert's solution. In addition, the invariants of f = 0 are used to analyze the singularity of a singular curve, since the number of the real inflection points of f = 0 depends on its singularity type. © 2003 Published by Elsevier Science B.V. http://hdl.handle.net/10722/89108 0167-83962017 Impact Factor: 1.5222015 SCImago Journal Rankings: 1.024 WOS:000182816000003 References in Scopus

DC FieldValueLanguage
dc.contributor.authorChen, Fen_HK
dc.contributor.authorWang, Wen_HK
dc.date.accessioned2010-09-06T09:52:29Z-
dc.date.available2010-09-06T09:52:29Z-
dc.date.issued2003en_HK
dc.identifier.citationComputer Aided Geometric Design, 2003, v. 20 n. 2, p. 101-117en_HK
dc.identifier.issn0167-8396en_HK
dc.identifier.urihttp://hdl.handle.net/10722/89108-
dc.description.abstractShape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature. A real inflection point is also required for transforming projectively a planar cubic algebraic curve to the normal form, in order to facilitate further analysis of the curve. However, the naive method for computing the inflection points of a planar cubic algebraic curve f = 0 by directly intersecting f = 0 and its Hessian curve H(f) = 0 requires solving a degree nine univariate polynomial equation, and thus is relatively inefficient. In this paper we present an algorithm for computing the real inflection points of a real planar cubic algebraic curve. The algorithm follows Hilbert's solution for computing the inflection points of a cubic algebraic curve in the complex projective plane. Hilbert's solution is based on invariant theory and requires solving only a quartic polynomial equation and several cubic polynomial equations. Through a detailed study with emphasis on the distinction between real and imaginary inflection points, we adapt Hilbert's solution to efficiently compute only the real inflection points of a cubic algebraic curve f = 0, without exhaustive but unnecessary search and root testing. To compute the real inflection points of f = 0, only two cubic polynomial equations need to be solved in our algorithm and it is unnecessary to solve numerically the quartic equation prescribed in Hilbert's solution. In addition, the invariants of f = 0 are used to analyze the singularity of a singular curve, since the number of the real inflection points of f = 0 depends on its singularity type. © 2003 Published by Elsevier Science B.V.en_HK
dc.languageengen_HK
dc.publisherElsevier BV. The Journal's web site is located at http://www.elsevier.com/locate/cagden_HK
dc.relation.ispartofComputer Aided Geometric Designen_HK
dc.subjectAlgebraic curveen_HK
dc.subjectHessian curveen_HK
dc.subjectInflection pointen_HK
dc.subjectInvarianten_HK
dc.subjectSingular pointen_HK
dc.titleComputing real inflection points of cubic algebraic curvesen_HK
dc.typeArticleen_HK
dc.identifier.openurlhttp://library.hku.hk:4550/resserv?sid=HKU:IR&issn=0167-8396&volume=20&issue=2&spage=101&epage=117&date=2003&atitle=Computing+real+inflection+points+of+cubic+algebraic+curvesen_HK
dc.identifier.emailWang, W:wenping@cs.hku.hken_HK
dc.identifier.authorityWang, W=rp00186en_HK
dc.identifier.doi10.1016/S0167-8396(03)00022-0en_HK
dc.identifier.scopuseid_2-s2.0-0037960142en_HK
dc.identifier.hkuros81582en_HK
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0037960142&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume20en_HK
dc.identifier.issue2en_HK
dc.identifier.spage101en_HK
dc.identifier.epage117en_HK
dc.identifier.isiWOS:000182816000003-
dc.publisher.placeNetherlandsen_HK
dc.identifier.scopusauthoridChen, F=7404908180en_HK
dc.identifier.scopusauthoridWang, W=35147101600en_HK