A.ProfChengyingCourseA.ProfChengyingCourse LectureSubdivisioncurvesandsurfaces細(xì)分曲面RepresentationofnaturalL-SystemL系統(tǒng)Particlesystem粒子系統(tǒng)Imlicitsurfaces隱式曲面WhatisSubdivisionisaprocessinwhichapoly-line/meshisrecursivelyrefinedinordertoachieveasmooth 1Cu廿bb盧盧凡Ja1a

口mf0,l201HowtoSubdivisionExtendingtoTypeofTwomaingroupsofApproximating-originalverticesareInterpolating–originalverticesareWhyFramefrom“Geri’sGame”byWhyLOD(levelofHistoryof‘80s,Loop,‘95,ReifIncreasedcomplexitytoRenderman==>Geri’sArbitraryTopology-nopatch,general/vertexScalability-levelofdetail,UniformityofRepresentation-meshedtoNumericalStability-goodforCodeBasicSubdivisionschemesareusuallydefinedondifferenttypesofpolygonalmeshesTriangularmeshes(三角網(wǎng)格 lmeshes(四邊形網(wǎng)格Tri/quadmeshes(混合網(wǎng)格Valenceofvertices頂點(diǎn)的價(jià)ThenumberofedgesadjacenttotheBasicRegularvertexandirregularVertexofvalence4inaquadmeshisregular;andirregularVertexofvalence6inatri.meshisregular;andirregularAteach RefineIncreasenumberofvertices y)*Meshverticesconvergetoalimit–AfterinfinitenumberofsubdivisionBasicEverysubdivisionmethodRulestocalculatelocationofnewverticesandoldiftheyareAmethodtorefinethemeshAschemealwaysconsistsof2mainRulestodeterminethegeometry(Thisvertexisat(x,y,z))oftheverticesinthenewmesh.Amethodtogeneratethetopology(Theseverticesareconnected)ofthenewmesh.Constructingthe TriangularWorksonlyfortriangleEverytriangle cedby4newTwokindsofnewGreenverticesassociatedwitholdBlackverticesassociatedwitholdReductiontoatriangularWhenmeshisnottriangular,preprocessingis，LoopSubdivisionTopologicrules:insertanewvertextoeachFirstFirstLoopSubdivisionGeometricrules:regularLoopSubdivisionGeometricrules:irregularLoopSubdivisionLoopsubdivision:TheLimit LimitsurfacesofLoop’ssubdivisionisC2almostFinitesetofsingularlocationswherethesurfaceInitial 1st 2nd LimitButterflyInterpolationNewblackverticesinheritlocationofoldNewgreenverticescomputedbyfollowing

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TheLimitLimitsurfacesofButterfly’ssubdivisionisC1almosteverywhereFinitesetofsingularlocationswherethesurfaceisC0.Example:Initial 1st 2nd LimitCatmull–Clarksurfacesaredefinedonameshofanarbitrarypolyhedron.jjnif inev1v2f14 n2v1

1j j

Topologicrules:insertavert.toeachedgeandeachface,theoldvert.is (butitisn’ttheThenewmeshwillconsistonlyofquadrilals,whichwon'tingeneralbenar.Thenewmeshwillgenerallylooksmootherthantheoldmesh.ComparisonofComparisonofComparisonofTheoreticalquestionsofImplementation:LoopUsehalf-edgetypedefstruct{floatx;floaty;floatz;_HEdge*_HVert*}

//從該點(diǎn)射出的其中一條typedefstruct{_HVert*_HEdge*_HFace*_HEdge*

//與該半邊相交的面的下_HVert*_HEdge*_HEdge*}

//細(xì)分得到的第1條半//細(xì)分得到的第2Face&typedefstruct{_HEdge* //與該面相交的其中一條}typedefstruct{CPtrListVert;CPtrListFace;}

頂點(diǎn)半邊AdaptiveNotallregionsofamodelneedtobeIdea:Usesomecriteriaandadaptivelysubdividemeshwhereneeded.Screensize(maketriangles<sizeofpixel)ViewDistancefromInviewCareful!Mustensurethat“cracks”aren’tSubdivisionSurfacesfor(Refinement)-ProgressiveGeometryAndreiKhodakovsky,PeterSchr?derandWimSweldens(SIGGRAPH2000)SubdivisionSurfaceSimplemethodfordescribingcomplexRelativelyeasytoArbitraryLocalGuaranteedcontinuity(C1orC2continuousBefitforanimation,rapid yticexpressionatirregularvertices奇異點(diǎn)處沒(méi)有解ItisdifficulttoconstructsurfacesofhigherorderSubdivisioncurvesandsurfaces細(xì)分曲面RepresentationofnaturalL-SystemL系統(tǒng)Particlesystem粒子系統(tǒng)Imlicitsurfaces隱式曲面RepresentationofnaturalModelingnaturalscenesisachallengeproblemMountains,trees,flowersandgrass,flame,cloud,smoke,fluid(山,樹,花草,火,云, ThreekindsofFractal分形L-Systembasedon rrules( Particlesystem粒子系統(tǒng)Mainfeaturesoffractal分形的特點(diǎn)Selfsimilarity自相似性)：localregionsofafractalaresimilartothefractalitselfInfinitesimaldetail無(wú)限小細(xì)節(jié))：AfractalalwaysexhibitsdetailsnomatterhowsmalltheregionisExample：KochsnowKochsnowflakecurve雪花曲線1/3原,RepeattheSierpinski三角形Mountaingenerationbased一維分–Let(xiyi)、(xi+1yi+1betwoendpoints,thenewlyinsertedpoint(xnew,ynew)iscalculatedas： 1(x 1(y )P(x

–Random(?)generatesarandomnumberin[0,1],P(?)isusedtocontrolthescale.Forexample，inthes-throundofi tion,wesetP(s)=2-s1DMountainbasedonfractal:Givenline

Movethemidpointalongy-axisbyarandomdistanceRepeattheprocess2Dmountain:Atriangleissplitintofour.(b)Themidpointofeachedgedisturbedalongy-Remarks:YouuseamorecomplexinitialrepeattherefinementusingsubdivisiontopologicMountainsurface:L-systems(Lindenmayersystem)ntAristidLindenmayer（ 爾）,17/11/1925–30/10/1989,Hungarian1968developedL-systemtomodelthebehaviourofcellsof L-systemsnowadaysarealsousedtomodelwhole L-system nt rrules：agr rrulere cesastringusinganewoneThe tionresultiscalledageneration一代Characterex nation：usegeometricprimitivesto cethecharactersTree:Characters：“A”,“B”,“[”“]”,“(”, rA→AA;2 B→A[B]AA(B)→Character nation(字符解釋“A”standsforastem;“B”isa“[]”:thebranchturnsleftwith“()”:turnright, (a) (b)1st (c)2stL-system:examplesfor ntIntroducemorecontrolinL-IllustratethecharacterofdifferentgenerationwithadifferentBranchesof(n+1)-thgenerationshouldbesmallerandshorterthanthatofthen-thgenerationFlowersandleavesareassignedtoterminationnodes終止節(jié)Usedifferent rrulesanddifferentTreesbasedonL-system:Furtherreading:TheAlgorithmicBeauty Oftenusedtosimulatefirefogsmokefluidand 隨時(shí)間變化的粒子粒子的運(yùn)動(dòng)由物理方法粒子被賦予生命generation,developmentandWILLIAMT.REEVES.ParticleSystemsm：ATechniqueforModelingaClassofFuzzyObjects.ACMTOG,2(2),1983:91-108GeneraldescriptionofparticleParticlesystemisadynamicThestepstogenerateaframeCreatenewparticlesandaddthemintotheAssignDeleteparticleswhohavediedMoveparticleaccordingtotheirRenderall1ParticleSpecifythenewparticlenumberofeachframe為每一幀指定新粒子數(shù)Npartsf=MeanPartsf+Rand()XVariancePartsfMeanPartf平均值；VarianceParts----變化范圍1≤rand(1Determinethenewparticlenumberaccordingtoscreensize(由屏幕大小指定粒子數(shù))Npartsf=(MeanPartsSAf+Rand()XVariancePartsSAf)X(按照這個(gè)方法新粒子數(shù)個(gè)數(shù)取決于屏幕面積2ParticleInitialInitial

InitialSize=MeanSize+Rand()XInitial 初始顏色I(xiàn)nitialInitial…

初 3Particlefield:gravitywindtension(壓力Collision(碰撞Spring(彈力Springsbetweenneighboringparticles…Dynamics（動(dòng)力學(xué)Newtonlaw:f=Gravity(加速度):a(t+△t)=gSpeed(速度):v(t△t)v(t)a(t)*tp(t+△t)=p(t)+v(t)*△t+4Particleextinction（粒 Forparticles,ifitslifetimeexceedsitslifecircle,weneedtoremoveitfromthesystem.ParticleRenderallparticlesasitsspecifiedFlameandwaterfall:粒子系統(tǒng)生成的火 粒子系統(tǒng)生成的瀑Naturalscene用其它方法可以模擬波浪、云和大氣、湍流、布料等 FurtherW.T.Reeves.ParticleSystems---aTechniqueforModelingaClassofFuzzyObjects,1983, TransactionsonGraphics(TOG),2(2)Real-timecloudsimulationandrendering,Siggraphcourse,2005.Subdivisioncurvesandsurfaces細(xì)分曲面RepresentationofnaturalL-Syste