計算機(jī)圖形學(xué)課件

計算計算機(jī)圖形ComputerDifferentialGeometryandDiscreteDifferentialDifferentialofCurvesand????DifferentialGeometryofaDifferentialGeometryofaIandIIFundamentalBendingDifferentialGeometryDifferentialGeometryofaDifferentialGeometryofDifferentialGeometryofaPointponthecurveatpDifferentialGeometryofaTangentTtotheDifferentialGeometryofaTangentTtothecurveatpTCuCuDifferentialGeometryofaNormalNandBinormalBtothecurveatDifferentialGeometryofaNormalNandBinormalBtothecurveatBpTCuuTCuuNN2CuBCuBCuCuuCuCuDifferentialGeometryofaCurvatureκDifferentialGeometryofaCurvatureκatu0andtheradiusρosculatingGeometricBpNDifferentialGeometryofaDifferentialGeometryofaCurvatureatu0isthecomponentof-NsalongTTComputingtheCurvatureofau1TNsusuuCCuuuTCs1sComputingtheCurvatureofau1TNsusuuCCuuuTCs1sCuuu ssuCu sCuCuComputingtheCurvatureofaCuCuuCNuuC CuuComputingtheCurvatureofaCuCuuCNuuC CuuuCuuCuCuuuCuCuCuuNC 2uuuCuCuuCuuCuCuuC 2uuCuCuu uuC2uuComputingtheCurvatureofaCuCuuCNuuC CuuuCuuComputingtheCurvatureofaCuCuuCNuuC CuuuCuuCuuCuCuuuCuCuNC 2uuuCuCuuCuuC 2uuCuCuu uuC2uuComputingtheCurvatureofaCuCuuuCuCuComputingtheCurvatureofaCuCuuuCuCuCuuCuuNC uuuCuCuu uuC2uuComputingtheCurvatureofaCuCuuuCuCuCuuCuuNCComputingtheCurvatureofaCuCuuuCuCuCuuCuuNC uuu CCuCuuuuCuCuu2CuCuCuuCuuCC uuuuCuCuuCuCuCuuCuuComputingtheCurvatureofaCuCuuuCuCuCuuCuuNC uuuComputingtheCurvatureofaCuCuuuCuCuCuuCuuNC uuu CCuCuuuuCuCuu2CuCuuCuCuCuCuuCuCuuCuCuCuCuCu3CuAAShortCurveCurve????DifferentialGeometryofaDifferentialGeometryofaIandIIFundamentalBendingDifferentialGeometryDifferentialGeometryofaDifferentialGeometryofDifferentialGeometryofaPointponthesurfaceatpDifferentialGeometryofaTangentDifferentialGeometryofaTangentSuintheuSu,SpuDifferentialGeometryofaTangentDifferentialGeometryofaTangentSvinthevSu,SpvDifferentialGeometryofaPlaneofDifferentialGeometryofaPlaneoftangentsTTpMetricofthesurface?STuvSSSuvMetricofthesurface?STuvSSSuvvSv vSvuvvFSvDifferentialGeometryofaNormalDifferentialGeometryofaNormalNNpTDifferentialGeometryofaDifferentialGeometryofaNormalNpTDifferentialGeometryofaNpDifferentialGeometryofaNpT1TTDifferentialGeometryofaTDifferentialGeometryofaTNpTTT1TTS NuTNTuvvvSNvS NuTNTuvvvSNv vvvuvNNvSNNvSMvNSv????DifferentialGeometryofaDifferentialGeometryofaIandIIFundamentalBendingChangeofChangeofpTangentPlaneofChangeofbθpaConstructanOrthonormalSSSa0ussSSbbsinSChangeofbθpaConstructanOrthonormalSSSa0ussSSbbsinSv ttbS0SS suu1aSSSabsinbt vvChangeofbθpFirstFundamentalaSS0SusAChangeofbθpFirstFundamentalaSS0SusAtISSSSSuvstvtabcosabtChangeofbTsutθvpaApointTexpressedin(u,v)andChangeofbTsutθvpaApointTexpressedin(u,v)andSutuvTvsSSvtSsSutStSv????DifferentialGeometryofaDifferentialGeometryofaIandIIFundamentalBendingκTisafunctionofdirectionbTθSNpuNvκTisafunctionofdirectionbTθSNpuNvauvvSvtA1IIA1tA1tSSStStbHowdoweanalyzetheκTθbHowdoweanalyzetheκTθpas A1IIA1TStSSEEigenanalysisof1?bEigenvalues={κ1,κ2}Eigenvectors=φθpA1 a AEEigenanalysisof1?bEigenvalues={κ1,κ2}Eigenvectors=φθpA1 a ATSSStEigendecompostionof0ss 22111St0tt2221s2E1S12stt21coscossinsincosSsinvbcossincoscoscossinsincosSsinvbcossincosEαcossin2φcosVθsinSpas?tcoscoscoscossin?sin IIsinsin0cossin2cos21sin2????DifferentialGeometryofaDifferentialGeometryofaIandIIFundamentalBendingWeingartenE1bWI A φSStA11tθS?SA1tA1pau t?11tAtSWeingartenE1bWI A φSStA11tθS?SA1tA1pau t?11tAtSVV111??AASVV0VA1t?A1tu2s21tv22211WeingartenWISSFM1NEGFEGLGMWeingartenWISSFM1NEGFEGLGMFN1EMENFMEGFGL2FMEN GL2FMEN24EGF2LNM22EGF2G EGF12TraceWGL2FM2EGF2M22WeingartenGLGMFN1WEGF2EMENWeingartenGLGMFN1WEGF2EMENFMGL2FMEN24EGF2LNM22EGF2GL2FMENIfκ1≠GL2FMEN24EGF2LNM2GLEN 2EMSuv1elseumbilic(κ1=κ2),choseorthogonalE2NSu????DifferentialGeometryofaDifferentialGeometryofaIandIIFundamentalBendingBending 2EB12S122M2GS4M2A2G Bending 2EB12S122M2GS4M2A2G 22B1222122 2SS4 2A22S22221 2M 4S212A42421S22MGBending 2EB12S122M2GS4M2A2G 22BBending 2EB12S122M2GS4M2A2G 22B1222122 2SS4 2A22S22221 2M 4S212A42421S22MGMinimizingSA=MinimizingSA1222MShortShortPrincipalMinPrincipalMinMaxSurfaceSurfaceDiscreteDiscreteDifferentialon三角網(wǎng)三角網(wǎng)格曲面的光滑性However,meshesareonlyHowever,meshesareonly?Meshesarepiecewiselinear–Infinitelycontinuousinsideeach–C0edgesandDiscreteDifferentialDiscreteDifferential?HowtoapplythetraditionalgeometryondiscretemeshNormal–EstimationofEstimationofDifferential?Approximatethe(unknown)Approximatethesurface&computecontinuousdifferentialmeasures(normal,curvature)ApproximatedifferentialmeasuresforContinuousContinuousQuadraticQuadraticQuadraticQuadraticApproximationQuadraticQuadraticApproximationQuadraticQuadraticApproximationOther?Cubic–andV.AOther?Cubic–andV.Adirectionvectors.ACMTransactionson23,1(2004),Implicitsurface–YutakaOhtakeetal.Multi-levelpartitionofunityimplicits.Siggraph2003.Many??DiscreteDiscreteNormal?NormalNormal?NormalestimationonDefinedforeachWeighted:faceareas,anglesat?WhathappenatMeanMeanMeanMeanCurvatureGaussianGaussian–MeanMeanCurvature–MeanMeanMEYERM.,DESBRUNM.,SCHR?DERP.,A.:Discretedifferential-geometryoperatorsfortriangulated2-manifolds.InVisualizationandMathematicsIII,HegeH.-C.,PolthierK.,(Eds.).Springer,2003,pp.35–58.(PDF)?????Feature????FeatureShaperecognitionAnyfeature-aware–Preservingsalientfeaturesin?–WhatarefeaturesonDifferentialDifferentialMeshMeshSurface:2DGraphin??Prosandv1.5-0.960751-v0.81-0.891238-v0.16-0.233535-v1.49-2.44325-v1.59-2.98815-v1.66-2.81016-v1.41-1.14861-v1-1.40023-v0.88-1.33122-v1.69-2.60816-v1.68-2.36516-…Differential(LaplaceDifferential(Laplace?Representlocaldetailateachsurface–betterdescribetheLineartransitionfromglobaltoUsefulforoperationsonsurfaceswheresurfacedetailsareimportant??What’sWhat’sareDetail=surface–smooth(surface)Smoothing=averaging??Differential?Differential?wivi averageofWhat’sthe?AbsoluteWhat’sthe?Absolutevi(xi,yi,zi?RelativejN(iwjvjviWeighting?wjWeighting?wj wj(cotcot??wjjjGeometric?DCsrepresentthelocaldetail/localTheGeometric?DCsrepresentthelocaldetail/localThesizeapproximatesthemeanvv len(vivδiidvN(ii1len(vvH(v)iiilen(MeshMeshLaplacianSmoothingLaplacianSmoothingLaplace(Umbrellann1n1nLaplace(Umbrellann1n1nQiiLQkQk