塩山幾何学を用いた ボロノイ図の解析

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1、AnalysisOfVoronoiDiagramsUsingTheGeometryofsaltmountainsRitsumeikanhighschoolMimuraTomohiroMiyazakiKosukeMurataKodaiWhensaltissprinkledontheboardwhichiscutintoaparticularshape,itcreatesmountainridgesThelinescreatedbythemountainridgescanexpressvariouscharacteristicApplicationofmodeling

2、biologybySaltMountainGeometricsWefoundthatVoronoidiagramscanbereproducedbysaltmountains.■Abstract■2Mr,Kurodasuggest“thegeometryofsalt”Whenalotofsaltispouredonaboardwhichiscutintoaparticularshape,itcreatesa“saltmountain.Wenamed“Geometryofsaltmountain”.1Whatisgeometryofsaltmountain31Wh

3、atisgeometryofsaltmountain4Whensomepointsareputlikethisonadiagram,aVoronoiDiagramisthediagramwhichseparatestheareasclosesttoeachpointfromtheotherpoints.2Whatisvoronoidiagram53themountainridgesformedbypouringsaltonvariouspolygons6Samedistanceincenter3-1Triangle73-2QuadrilateralsandPe

4、ntagons83-2ExaminationofQuadrilaterals△ＡＢＥの内心点△ＡＢＥの傍心点93-2ExaminationofPentagons103-2四角形　五角形結果内心円が書ける四角形では頂点が一つになった113-3ConcaveQuadrilateralsandPentagons12ThereasonofappearingcurvelineisthattherearedifferentshortestlinefromaconcavepointPointEissamedistancetolineｌandATherewerecurvelin

5、es.3-3Examination133-3Examination143-4acircleboardwithahole153-4Examination16ED＝EACE＋BE＝CE＋EA＋AB＝CE＋ED＋AB＝CD＋AB＝（bigcircle’sradius）＋（smallcircle’sradius）＝Constant3-5QuadraticCurves17p＜PQp＞PQ3-5Examination18dThusthemountainridgesaredisappearedatp<1/2.Ifp＞１/2,Ifp＜1/2,theminimumTosolvedw

6、hichismakeup(0,p)ony-axisandQony=x23-5Examination193-6OneHole203-6TwoHoles214applicationstoVoronoiDiagrams224-1Flowcharting234-2SimulationoftheprogramComparetosaltmountain244-2SimulationoftheprogramComparetosaltmountain25WeightedVoronoiDiagramsareanextensionofVoronoiDiagrams.d(x,p(i)

7、)=d(p(i))-w(i)4-3AdditivelyweightedVoronoiDiagrams26saltmountainscouldreproducethisbyreplacingweightwiththeradiusofthehole.thismeanweight=radius4-4Relationwithweightandradius274-4Relationwithweightandradius284-5SimulationoftheprogramComparetosaltmountain294-5Simulationoftheprogram

8、Compa