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1、数理解析研究所講究録1378巻2004年138-153138AnAlgebraicFoundationforObject-OrientedEuclideanGeometryLeoDorstandDanielFontijneInformaticsInstitute,UniversityofAmsterdam,TheNetherlandsAbstractTheconformalmodelofEuclideangeometryinGeometricAlgebraprovidesacompactwaytocharacteriz
2、eEuclideanobjectssuchasspheres,planes,circles,lines,etc.asblades.Thealgebraicstructureofthemodelprovidesa‘grammar’fortheseobjectsandtheirrelationships.Inthisratherinformalpaperweexplorethisgrammar,developinganewgeO-metricintuitiontouseiteffectively.Thisresultsint
3、heidentificationoftwoimportantconstructionproducts,theknownmeetandthenewplunge.Theseprovidecompactspecificationtechniquestoparametrizeoperatorsandobjectsdirectlyintermsofotherobjects.1Introduction1.1EuclideanPrimitivesasSubspacesAnelegantmodelforEuclideangeometryw
4、asintroducedrecently[8],calledthe‘conformalmodel’(sinceitcansupportconformaltransformationsaswell,al-thoughwedonotusethoseinthispaper).TheideabehindtheconformalmodelistoembedtheEuclideanspace$E^{n}$intotheMinkowskispace$mathrm{m}^{n+1,1}$,andtheEuclideanmetrici
5、ntotheinnerproductofthatMinkowskispace.Subspacesof$mathrm{m}^{n+1,1}$arebladesinitsgeometricalgebra,andeasilyinterpretableasprimitiveobjectsinEn.TheoperatorsofgeometricalgebrathenorganizetheEuclideangeometryalgebraically,andthisresultsinuseful‘datatypes’forelem
6、entaryge-ometrywithwell-understoodrelationships.Thistechniqueismetric,andconsid-erablyextendsthecommon,non-metric,homogeneouscoordinatemethodsformodelingEuclideangeometry.Extensiveintroductionstothis‘conformalmodelofEuclideangeometry’areavailable[2].Herewejustbr
7、ieflyrepeatthemainpointsrequiredforworkingwithit.Thepaperprovidesamoreintuitiveunderstandingofthemodelingmethodanditsadvantages.Ifyouwouldliketoplaywithit,werecommendourinteractivetutorialrunninginGAViewer[2].1391.2TheconformalmodelinbriefFirst,weextendtheEuclide
8、anspacewithapointatinfinity.Werepresentthisasaparticularvectorof$mathrm{m}^{n+1,1}$,andinthispaperwedenotethisvectorbythesymbol$infty$.WerepresentageneralEuclideanpo
