The differential Geometry of parametric primitives

《The differential Geometry of parametric primitives》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

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¶xúë¶v¶v¶vûë¶vûTheHessianisatensorofsecondpartialderiva

4、tives:éé¶2x¶2y¶2zùé¶2x¶2y¶2zùù2êê222úêúú¶(x,y,z)ë¶u¶u¶uûë¶u¶v¶u¶v¶u¶vûH==êú222222¶(u,v)¶(u,v)êé¶x¶y¶zùé¶x¶y¶zùúêêúê222úúëë¶v¶u¶v¶u¶v¶uûë¶v¶v¶vûû22é¶x¶xùê2ú¶u¶u¶v=êú22¶x¶xêúëê¶v¶u¶v2úûThefirstfundamentalformisdefinedas:é¶x¶x¶x¶xù··êút¶u¶u¶u¶vG=JJ=êú¶x¶x¶x¶xê··úë¶v¶u¶v¶vûA

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·ëê¶v¶u¶v2úûThenormalcurvatureisdefinedtobepositiveacurveuonthesurfaceturnstowardthepositivedirectionofthesurfacenormalby:uDúuútk=ntuGúuúThedeviation(inthenormaldirection)fromthetangentplaneofthesurface,givenadifferentialdisplacementofu

8、úis:txúú·n=uDúuúAppleComputer,Inc.MediaTechnology:ComputerGraphicsPage2TurkowskiTheDifferentialGeometry