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1、CircleandsphereblendingwithconformalgeometricalgebraChrisDoran1AstrophysicsGroup,CavendishLaboratory,MadingleyRoad,CambridgeCB30HE,UK.AbstractBlendingschemesbasedoncirclesprovidesmooth‘fair’interpolationsbetweenseriesofpoints.Herewedemonstrateasimple,robustsetofalgorithmsforperformingcircleblends
2、forarangeofcases.AnarbitrarylevelofG-continuitycanbeachievedbysimplealterationstotheunderly-ingparameterisation.Ourmethodexploitsthecomputationalframeworkprovidedbyconformalgeometricalgebra.Thisemploysafive-dimensionalrepresentationofpointsinspace,incontrasttothefour-dimensionalrep-resentationtypi
3、callyusedinprojectivegeometry.Theadvantageoftheconformalschemeisthatstraightlinesandcirclesaretreatedinasingle,unifiedframework.Asafurtherillustrationofthepoweroftheconformalframework,thebasicideaisextendedtothecaseofsphereblendingtointerpolateoverasurface.Keywords:spline,geometry,geometricalgebra
4、,conformal1IntroductionInarangeofapplicationsweoftenseekcurvesandsurfacesthathaveanaesthet-icallypleasing‘roundedness’tothem.Onewaytomakethisconceptconcreteisthroughlookingforglobally-optimised‘minimumvariationcurves’[1].Thephilosophybehindthisideaisstraightforward.Weusuallyprefercurvesthatareclo
5、setocircularovercurveswithsharpturns.Thisisparticularlytruewhendesigningcameratrajectories,wheresuddenchangesincurvaturecanhaveaverydisorientingeffect.Circularpathsarecharacterisedbyhavingconstantcurvature,soanaturalideainforminginterpolationsbetweencontrolpointsistofindacurvethatminimisesthetotalc
6、hangeincurvature.Theproblemwithsuchastrategyisthatthesecurvescanbeextremelyhardtocompute.Ifoneadoptedavariationalstrategy,withendpointconditions,theequationsforthecurvecanbeashighasfifthorderandareevenmoredifficulttotreatthanarXiv:cs/0310017v1[cs.CG]9Oct2003thoseofelasticity.Suchequationscanonlybeso
7、lvednumericallyanddonothavestraightforward,controllable,analyticsolutions.Theproblemisevenmoreacuteifmultiplecontrolpointsareinvolved,asevennumericalcomputationcanbeextremelydifficult.Amorestraightforward,localschemethat
