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1、2.GeneralizedHomogeneousCoordinatesforComputationalGeometryyHONGBOLI,DAVIDHESTENESDepartmentofPhysicsandAstronomyArizonaStateUniversityTempe,AZ85287-1504,USAALYNROCKWOODPowerTakeO?Software,Inc.18375HighlandEstatesDr.ColoradoSprings,CO80908,USA2.1IntroductionThestandardalgebraicmodelforEucl
2、ideanspaceEnisann-dimensionalrealvectorspaceRnor,equivalently,asetofrealcoordinates.Onetroublewiththismodelisthat,algebraically,theoriginisadistinguishedelement,whereasallthepointsofEnareidentical.Thisdeˉciencyinthevectorspacemodelwascorrectedearlyinthe19thcenturybyremovingtheoriginfromthe
3、planeandplacingitonedimensionhigher.Formally,thatwasdonebyintroducinghomogeneouscoordinates[H91].ThevectorspacemodelalsolacksadequaterepresentationforEuclideanpointsorlinesatinˉnity.WesolvebothproblemsherewithanewmodelforEnemployingthetoolsofgeometricalgebra.WecallitthehomogeneousmodelofEn
4、.Our10ewmodel"hasitsoriginsintheworkofF.A.Wachter(1792{1817),astudentofGauss.HeshowedthatacertaintypeofsurfaceinhyperbolicgeometryknownasahorosphereismetricallyequivalenttoEuclideanspace,soitconstitutesanon-EuclideanmodelofEuclideangeometry.Withoutknowl-edgeofthismodel,thetechniqueofcomfor
5、malandprojectivesplitsneededtoincorporateitintogeometricalgebraweredevelopedbyHestenesin[H91].Theconformalsplitwasdevelopedtolinearizetheconformalgroupandsimplifytheconnectiontoitsspinrepresentation.Theprojectivesplitwasdevelopedtoin-corporatealltheadvantagesofhomogeneouscoordinatesinaoord
6、inate-free"representationofgeometricalpointsbyvectors.AndraesDressandTimothyHavel[DH93]recognizedtherelationoftheconformalsplittoWachter'smodelaswellastoclassicalworkondistancegeometrybyMenger[M31],Blumenthal[B53,61]andSeidel[S52,55].Theyalsostressedconnectionstoclassicalinvarainttheory,fo
7、rwhichthebasicshavebeenincorporatedintogeometricalgebrain[HZ91]and[HS84].ThepresentworksynthesizesallthesedevelopmentsandintegratesconformalandprojectivesplitsintoapowerfulalgebraicformalismforrepresentingandmanipulatingyThisworkhasbeenpartiallysupportedbyNSFG
