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1、THEDIFFERENTIALGEOMETRYOFPARAMETRICPRIMITIVESKenTurkowskiMediaTechnologies:GraphicsSoftwareAdvancedTechnologyGroupAppleComputer,Inc.(DraftFriday,May18,1990)Abstract:Wederivetheexpressionsforfirstandsecondderivatives,normal,metricmatrixandcurvaturematrixforspheres,cones,c
2、ylinders,andtori.26January1990AppleTechnicalReportNo.KT-23TurkowskiTheDifferentialGeometryofParametricPrimitives26January1990TheDifferentialGeometryofParametricPrimitivesKenTurkowski26January1990DifferentialPropertiesofParametricSurfacesAparametricsurfaceisafunction:x=F(
3、u)wherex=[xyz]isapointinaffine3-space,andu=[uv]isapointinaffine2-space.TheJacobianmatrixisamatrixofpartialderivativesthatrelatechangesinuandvtochangesinx,y,andz:é?x?y?zùé?xùêúêú?(x,y,z)?u?u?u?uJ==êú=êú?(u,v)ê?x?y?zúê?xú??v?v?v???v?TheHessianisatensorofsecondpartialderiva
4、tives:éé?2x?2y?2zùé?2x?2y?2zùù2êê222úêúú?(x,y,z)??u?u?u???u?v?u?v?u?v?H==êú222222?(u,v)?(u,v)êé?x?y?zùé?x?y?zùúêêúê222úú???v?u?v?u?v?u???v?v?v??22é?x?xùê2ú?u?u?v=êú22?x?xêú?ê?v?u?v2ú?Thefirstfundamentalformisdefinedas:é?x?x?x?xù··êút?u?u?u?vG=JJ=êú?x?x?x?xê··ú??v?u?v?v?A
5、ppleComputer,Inc.MediaTechnology:ComputerGraphicsPage1TurkowskiTheDifferentialGeometryofParametricPrimitives26January1990andestablishesametricofdifferentiallength:2t(dx)=(du)G(du)sothatthearclengthofacurvesegment,u=u(t),t
6、òt(uGúuú)dt0dt000Thedifferentialsurfaceareaenclosedbythedifferentialparallelogram(du,dv)isapproximately:1dS?(G)2dudvsothattheareaofaregionofthesurfacecorrespondingtoaregionRintheu-vplaneis:12S=òò(G)dudvRThesecondfundamentalmatrixmeasuresnormalcurvature,andisgivenby:22é?x
7、?xùên·2n·ú?u?u?vD=n·H=ê22ú?x?xêún·n·?ê?v?u?v2ú?Thenormalcurvatureisdefinedtobepositiveacurveuonthesurfaceturnstowardthepositivedirectionofthesurfacenormalby:uDúuútk=ntuGúuúThedeviation(inthenormaldirection)fromthetangentplaneofthesurface,givenadifferentialdisplacementofu
8、úis:txúú·n=uDúuúAppleComputer,Inc.MediaTechnology:ComputerGraphicsPage2TurkowskiTheDifferentialGeometry
