关于常曲率曲面上测地线的密度公式与Crofton公式的注记

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1、西南大学硕士学位论文摘要其中n表示e和C交点的个数．．枣理譬．8．设K为单位球面上周长为L的有界凸集，则大圆集{c：CNK≠o)的不变测度为：上恻毋dc=L．定理5．2．设F为双曲平面上长为L的测地线段，则测地线集{C：CnF≠D)的不变测度为：上州。dc=2L．·定理5．3．设F为双曲平面上闭的测地多边形，其周长为L，则测地线集{c：CNF≠D)的不变测度为：上似。礼dc=2三．其中他表示F和C交点的个数。定理5．4．设F为双曲平面上周长为L的有界凸集，则测地线集{c：CNF≠0]．的不变测度为：L。一L关键词：

2、常曲率曲面密度测地线Crofton公式Blaschke公式西南大学硕士学位论文ABSZF沮CTRemarksAbouttheDensityoftheGeodesicAndCroftonFormulaontheSurfaceofConstantCurvatureMajor：IntegralGeomeryName：DengLingfangSupervisor：ProfessorZhouJiazuABSTRACTInthispaper，wemainlyinvestigatethreeproblems．Firstly,we

3、inVestigatethedensityontheEuclideanplaneR2．Weobtainseveraldensityformsoftheline．Accordingtothedensityformsoftheline，wegettheCroftonformulaaboutthemeasureofthelineswhichintersecttheconvexsetandBlaschkeformulaontheEuclideanplane．Thenweextendthedensityfbrmsofthel

4、inetotheunitsphere．Thatisinvestigatingtheformsofthegreatcircle’Sdensity．WeobtaintheCroftonformulaaboutthemeasureofthegreatcircleswhichintersecttheconvexsetandBlaschkeformulaontheunitsphere．Finally,weconsiderthedensityofthegeodesiconthehyperbolicplaneanalogousl

5、y．WeextendtheCroftonformulaaboutthemeasureofthelineswhichintersecttheconvexsetandBlaschkeformulatothehyperbolicplane．Themainresultsobtainedcanbesummarizedasfollows：Theorem4．2．LetCoandGbethegreatcirclesontheplaneyOzandxOy．DistheintersectionpointofCoandQ．Ifthegr

6、eatcircleCisdeterminedbyDAwhichiSnotedtandtheangleQbetweenCandG，thendC=sinadaAdt．Theorem4．3．LetQ1betheintersectionpointofthegreatcirclesCandCOJ_，‘、、ontheunitsphere，Q1Disnoteds．IfthegreatcircleCisdeterminedbysandt，thendC=sinsds八出．Theorem4．4．Let(z，Y，名)bethecoord

7、inateofthepoleQ，thendC：：—dxA—dy．Znl西南大学硕士学位论文ABSTRACTTheorem4．5．LetthegreatcircleCcutsacurveC7inapointP，妒istheanglebetweenC7andC，8istheabscissaofthepointPuponC7，thendC=sin妒d妒Ads．Theorem4．6．LetCbeageodesiclinewhichlengthisLontheunitsphere，thenthemeasureofsetsof

8、greatcircles．[C：Cn0≠D)isf，dC=2L．Jcno#oTheorem4．7．LetCbeaclosedpolygonwhichlengthisLontheunitsphere，thenthemeasureofsetsofgreatcircles{c：Cn0≠西)isl／一ndC=2L．Jcec≠O。蹄五erenisthenumberof