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#### Article: The μ-basis of a planar rational curve - Properties and computation

Title The μ-basis of a planar rational curve - Properties and computation Chen, FWang, W Μ-BasisImplicitizationMoving LineRational CurveSyzygy Module 2002 Academic Press. The Journal's web site is located at http://www.elsevier.com/locate/gmod Graphical Models, 2002, v. 64 n. 6, p. 368-381 How to Cite? A moving line L(x, y; t) = 0 is a family of lines with one parameter t in a plane. A moving line L(x, y; t) = 0 is said to follow a rational curve P(t) if the point P(t0) is on the line L(x, y; t0) = 0 for any parameter value t0. A μ-basis of a rational curve P(t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P(t), which is the syzygy module of P(t). The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3,6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P(t), and has O(n2) time complexity, where n is the degree of P(t). We show that the new algorithm is more efficient than the fastest previous algorithm [7]. © 2003 Elsevier Science (USA). All rights reserved. http://hdl.handle.net/10722/152199 1524-07032015 Impact Factor: 0.8212015 SCImago Journal Rankings: 0.443 WOS:000182649000002 References in Scopus

DC FieldValueLanguage
dc.contributor.authorChen, Fen_US
dc.contributor.authorWang, Wen_US
dc.date.accessioned2012-06-26T06:36:29Z-
dc.date.available2012-06-26T06:36:29Z-
dc.date.issued2002en_US
dc.identifier.citationGraphical Models, 2002, v. 64 n. 6, p. 368-381en_US
dc.identifier.issn1524-0703en_US
dc.identifier.urihttp://hdl.handle.net/10722/152199-
dc.description.abstractA moving line L(x, y; t) = 0 is a family of lines with one parameter t in a plane. A moving line L(x, y; t) = 0 is said to follow a rational curve P(t) if the point P(t0) is on the line L(x, y; t0) = 0 for any parameter value t0. A μ-basis of a rational curve P(t) is a pair of lowest degree moving lines that constitute a basis of the module formed by all the moving lines following P(t), which is the syzygy module of P(t). The study of moving lines, especially the μ-basis, has recently led to an efficient method, called the moving line method, for computing the implicit equation of a rational curve [3,6]. In this paper, we present properties and equivalent definitions of a μ-basis of a planar rational curve. Several of these properties and definitions are new, and they help to clarify an earlier definition of the μ-basis [3]. Furthermore, based on some of these newly established properties, an efficient algorithm is presented to compute a μ-basis of a planar rational curve. This algorithm applies vector elimination to the moving line module of P(t), and has O(n2) time complexity, where n is the degree of P(t). We show that the new algorithm is more efficient than the fastest previous algorithm [7]. © 2003 Elsevier Science (USA). All rights reserved.en_US
dc.languageengen_US
dc.publisherAcademic Press. The Journal's web site is located at http://www.elsevier.com/locate/gmoden_US
dc.relation.ispartofGraphical Modelsen_US
dc.subjectΜ-Basisen_US
dc.subjectImplicitizationen_US
dc.subjectMoving Lineen_US
dc.subjectRational Curveen_US
dc.subjectSyzygy Moduleen_US
dc.titleThe μ-basis of a planar rational curve - Properties and computationen_US
dc.typeArticleen_US
dc.identifier.emailWang, W:wenping@cs.hku.hken_US
dc.identifier.authorityWang, W=rp00186en_US
dc.identifier.doi10.1016/S1077-3169(02)00017-5en_US
dc.identifier.scopuseid_2-s2.0-0012367274en_US
dc.identifier.hkuros81577-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-0012367274&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume64en_US
dc.identifier.issue6en_US
dc.identifier.spage368en_US
dc.identifier.epage381en_US
dc.identifier.isiWOS:000182649000002-
dc.publisher.placeUnited Statesen_US
dc.identifier.scopusauthoridChen, F=7404908180en_US
dc.identifier.scopusauthoridWang, W=35147101600en_US