計(jì)算機(jī)圖形學(xué)課件cgmeshes

計(jì)算計(jì)算機(jī)圖形Computer三維物體的表?二三維物體的表?二次球面、拋物面圓柱錐????復(fù)雜數(shù)學(xué)表函數(shù)數(shù)學(xué)表函數(shù)參數(shù)表如何畫曲面？如何畫曲面？--離函數(shù)參數(shù)表多面體網(wǎng)多面體網(wǎng)格模?每個面為平面多邊WhyWhymeshSourcesSourcesof3DTwoTwomaintypesof正多正多面Euler對于Euler對于簡單多面體V+F–E頂點(diǎn)數(shù)：V,面數(shù)F邊數(shù)例如，立方體：V=8,F=6,V+F–E=2+H–??Euler公式VEuler公式V=16,F=16,E=32,H=G=V=24,F=15,E=36,H=G=如何表達(dá)模型的表面如何表達(dá)模型的表面?3D空間的3D3D物體的數(shù)字表3D物體的數(shù)字表達(dá)：點(diǎn)和三角?點(diǎn)三維坐標(biāo)三角網(wǎng)格(Triangular?三角網(wǎng)格(Triangular?點(diǎn)?線?面如何存儲如何存儲?網(wǎng)格模型數(shù)據(jù)格WavefrontOBJFile?WavefrontOBJFile?StartwithcharStartwithcharIndicesofitsverticesinthe??–Normal,texturecoordinates,material,v1.00.0v0.01.0v0.0-1.0v0.00.0f12f14f32f13Demo:Demo:Mesh2D2DDelaunayandMeshBorisN.BorisN.??RussianMarch15,1890-JulyIntroduceDelaunaytriangulationin?GeorgyF.GeorgyF.??RussianApril28,1868-20,PropertiesofPropertiesofDT?Emptysphereproperty:nopointsinsidecircum-sphereofany–DelaunayPropertiesofDTPropertiesofDT?DTmaximizesthesmallest–[Lawson1977]and[SibsonPropertiesofPropertiesofDT?Convexhull:unionofallPropertiesofDTPropertiesofDT?DTmaximizesthearithmeticmeanoftheradiusinscribedcirclesofthe[LambertDTminimizesroughness(theDirichletenergyofanypiecewise-linearscalarfunction)–[RippaDTminimizesthemaximumcontainingradius(theradiusofthesmallestspherecontainingthesimplex)[AzevedoandSimpson1989],[Rajan??PropertiesofPropertiesofDT?TheDTind-spacesistheprojectionthepointsofconvexhullontoa(d+1)-dimensional–[BrownPropertiesofPropertiesofDT?DTminimizesthespectrumoftheLaplacian(spectral–[Chenetal.Edge[SibsonEdge[SibsonStartwithany1.findanytwoadjacenttrianglesthatformaconvexquadrilateralthatdoesnotsatisfyemptyspherecondition2.swapthediagonalofthequadrilateraltobeaDeluanytriangulationofthatfourpoints3.repeatstep1,2until?Convergence?IsitpossibletoendwithaninfiniteMesh?GivenMesh?Givenafixedpointset,DelaunaywilltrytomakethetriangulationmoreregularandthusisconsideredasaMesh?WhatMesh?Whatdowemeana“good”MinimalAspect/radiusMean–Itisnoteasytodefineauniversalmeshqualityacceptablebyeveryone.Buteveryoneagreesonthe"best"simplex:equilateraltriangleandtetrahedra.?DTisDTisnotnecessaryagoodDTonlyoptimizetheconnectivitywhenpointsarefixed.distributionofpointsismoreimportantforagoodCentroidalVoronoiCentroidalVoronoi?Definition:TheVTisacentroidaltessellation(CVT),ifeachseedwiththecentroidofitsVoronoiLloyd????ConstructtheLloyd????ConstructtheVTassociatedwiththeComputethecentroidsoftheVoronoiregionsMovethepointstothecentroidsIterateuntilDefinitions&StandardGraphCBADLJIKEStandardGraphCBADLJIKEHFGGraphGraphPlanarPlanarPlanarPlanarGraphsandTopologyTopology?Delaunay?DelaunayTriangulationvs.VoronoiMeshMeshDataUsesofMesh?–UsesofMesh?–TriangleGeometry?–––WhataretheverticesoffaceArevertices#iand#jWhichfacesareadjacenttoface?Geometry–––Remove/addaMeshStoringMeshStoringMeshData?StorageofgenericHardtoimplement?StoringMeshDataStoringMeshData?How“good”isadataSpace???Timetoconstruct-TimetoansweraTimetoperformanoperation(updatethedataTrade-offbetweentimeandDefineaMeshDefineaMesh?Howdovertices?DefineaDefineaMesh????ListofVertex-3DMesh?Surface&3DMesh?Surface&material––––MaterialTexturecoordinates?Rendering–––RenderingGeneralUsedMeshGeneralUsedMesh?Generalusedmesh––––––3DMax(*.max,*.3ds)Inventor(*.iv)?WavefrontOBJFile?WavefrontOBJFile?StartwithcharStartwithcharIndicesofitsverticesinthe??–Normal,texturecoordinates,material,v1.00.0v0.01.0v0.0-1.0v0.00.0f12f14f32f13ListListofListofListof?ListofPositionListofTripletsofpointerstofaceverticesWhataretheverticesoffaceAnsweredinO(1)-checkingthirdAreverticesiandjApassoverallfacesisnecessary–NOT??ListListofFaces–ListofListofFaces–?Convenientandefficient(memoryCanrepresentnon-manifoldToosimple-notenoughinformationonrelationsbetweenvertices&faces?AdjacencyAdjacencyAdjacencyMatrixAdjacencyMatrix–ViewmeshasconnectedGivennverticesbuildn*nmatrixofadjacencyEntry(i,j)isTRUEvalueifverticesiandjareGeometriclistofvertexAddlistoftripletsofvertexindices????AdjacencyAdjacencyMatrix–AdjacencyMatrix–AdjacencyMatrix–?WhataretheverticesoffaceO(1)–checkingthirdtripletofAreverticesiandjO(1)-checkingadjacencymatrixatlocationWhichfacesareadjacenttovertexFullpassonallfacesis??AdjacencyMatrixAdjacencyMatrix–?InformationonvertexStoresnon-manifoldConnectsfacestotheirvertices,BUTNOconnectionbetweenvertexanditsface?Doubly-ConnectedDoubly-ConnectedEdgeRecordRecordforeachface,edgeandGeometricHalf-Edge??DCELDCELDCELDCEL–DCELDCEL–ExampleDCEL–DCEL–?AllqueriesinO(1)AlloperationsareO(1)Representson