CADCAM 原理与应用（英文版）课件 Part 3 Reverse Engineering、Part 4 surface modeling、Part 5 3D printing

主讲人：康兰CAD/CAM:PrinciplesandApplicationsCAD/CAM原理与应用CollegeofMechanicalandElectricalEngineering,HohaiUniversity（河海大学机电工程学院）

Part3:ReverseEngineering3.1Whatisreverseengineering(RE)?Contents3.2WhyREisneeded?3.5

ProcessofRE?3.3REhardware3.4REsoftware3.6REmodelingmethods3.7REapplications3.1WhatisReverseEngineering(RE)?WhatisReverseEngineering(RE)?Broadlyspeaking,reverseengineeringistheprocessofdiscoveringthetechnologicalprinciplesofadevice,object,orsystemthroughanalysisofitsstructure,function,andoperation.Inthiscourse,REreferstotheprocessofretrievingtheoriginalgeometryfromanexistingpartbyscanning,modifyingandrebuilding.3.2WhyisREneeded?WhyisREneeded?Theoriginalmanufacturernolongerexistsortheydon’tproducetheproduct,butacustomerneedstheproduct.Theoriginalproductdesigndocumentationhasbeenlostorneverexisted.Inspectionand/orqualitycontrol–comparingafabricatedparttoitsCADmodelortoastandarditem.Creating3Ddatafromamodelorsculptureforanimationingamesandmovies.Creating3Ddatafromanindividualmodelorsculpturetocreate,scale,orreproduceartwork.Generatingdatatocreatedentalorsurgicalprosthetics,tissueengineeredbodyparts,orforsurgicalplanning.…………3.2WhyREisneeded?3.3REhardwareREhardwareContactScannersCoordinateMeasuringMachines(CMM)PortableMeasuringArmsGantryCMM3.3REhardwareNoncontactscanners3.3REhardwareScanningprocess3.3REhardwareAboutrebuildingthemodelofapartfromscanninganassembly.Anexample:3.3REHardwareREsoftwareImagewareGeomagicStudioCopyCADRapidFormMost3DsoftwarealsoprovidesfunctionstoacceptandeditdatafromREsoftwareordirectlyfrom3Dscanners.3.4RESoftware3.5ProcessofREREModelingMethods3.6REModelingMethodsRebuildmodelbyfittingsurfacestoidentifiedfeatures3.6REModelingMethodsThekeystepforthismethodistosegmentpointclouddataintosmallregion,eachregionhassameattribute.Segmentingdatamethod:Example1:Setcertainconditions,suchasradiusofcurvature,thensegmentdatabydetectedsharpedges.3.6REModelingMethodsExample2:"regiongrowing"approach3.6REModelingMethodsExample3:Slicingthedata3.6REModelingMethodsExample3:Multi-directionalslicingManyways…..,moreresearchworkonthistopicstillcontinue.3.6REmodelingmethodsREApplicationsInaerospaceandautomotiveindustries.3.7REApplicationsMoulddesignManufacturingerrorinspectionBiomedicalapplicationHumanbodyshapemeasurementandrebuilding3.7REApplicationsRestoration/RepairofculturalrelicsMeasurementJointdataResearchershadtriedtorebuildthemodeloftheSaintVojtechlocatedonCharlesbridgeinPragueofCzechRepublic.3.7REApplicationsFinalcloudsofpoints3.7REApplicationsDigitalmodelofMaogaoCaves：3.7REApplications备注：网上有人说用三维扫描重建的方法完成的莫高窟三维模型图，但缺发细节及完成单位等详细信息，是否真实有待考证，但可从此例看出RE技术在文物保持方面的应用前景。主讲人：康兰CAD/CAM:PrinciplesandApplicationsCAD/CAM原理与应用CollegeofMechanicalandElectricalEngineering,HohaiUniversity（河海大学机电工程学院）

Part4:SurfaceModeling4.1IntroductiontoSurfaceModel4.5B-spline4.7NURBSContents4.2Mathematicalrepresentationsofcurves4.3IntroductiontoSplines4.4BezierCurves/Surfaces4.6IntuitiveUnderstandingofB-splines4.8ContinuityofCurves/Surfaces4.9SurfacesBasedonTriangleMeshes4.1IntroductiontoSurfaceModelSolidModelRepresentations:WireframeModelSolidModelSurfaceModelSurfacemodelsare:usedin3Danimationforgamesandotherpresentations.usedinCADforillustrations,architecturalrenderings.usedformodelingsurfacesofengineeringcomponentsfordesign,simulation,documentation,qualitycontroletc.4.1IntroductiontoSurfaceModelTypesofcurves/surfaces:irregularcurves/surfaces(freeformsurfaces)regularcurves/surfacesHowtodescribeirregularcurves/surfacesmathematically?4.1IntroductiontoSurfaceModel1.explicit,wherethecurveisgivenasthegraphofafunction.2.Implicit,astheglobalalgebraicequations.3.Parametric,associatedwithavectorfunctionofoneparameter.

4.2MathematicalrepresentationsofcurvesParametricrepresentationismostwidelyusedbyCADapplication.Whatareparametersandparameterization?Parametersaretheuniquenumericvalues(likecoordinates)ofpointsonacurveorsurface.Theapplicationofusingparameterstodecidethespecificpointsalongthecurveiscalledparameterization.Parametersletyourefertospecificpointsalongthelengthofacurve.Thehighertheparameter,thefurtheristhepointalongthecurve.4.2MathematicalrepresentationsofcurvesExamples:parametricrepresentationof

（1）astraightline4.2Mathematicalrepresentationsofcurves(2)acircleofradius1,centeredattheorigin(unitcircle)

Whyareparametricfunctionsneeded?4.2Mathematicalrepresentationsofcurves(3)asphereofradius1,centeredattheorigin

4.2MathematicalrepresentationsofcurvesDegreeofacurvePolynomialshavethegeneralform:y=a+bx+cx2+dx3+…Thedegreeofapolynomialcorrespondswiththehighestcoefficientthatisnon-zero.Itisthelargestexponentinaequation.4.2MathematicalrepresentationsofcurvesE.g:y=a+bx+cx2+dx3+…4.2MathematicalrepresentationsofcurvesExample:Y=2Degree1:Linear,bishighestnon-zerocoefficient.

Aline,uniquelydefinedbytwopoints.Degree2:Quadratic,cishighestnon-zerocoefficient.

Aparabola,uniquelydefinedbythreepoints.Degree0:Constant,onlyaisnon-zero.4.2MathematicalrepresentationsofcurvesDegree3:Cubic,dishighestnon-zerocoefficient.

Thedegreethreepolynomial,knownasacubicpolynomial,istheonethatismosttypicallychosenforconstructingsmoothcurvesincomputergraphics.Itisusedbecause:1.itisthelowestdegreepolynomialthatcansupportaninflection,sowecanmakeinterestingcurves.2.itisverywellbehavednumerically—thatmeansthatthecurveswillusuallybesmooth.4.3IntroductiontoSplinesoriginallydevelopedforship-buildingAbithistory:4.3IntroductiontoSplinesLong,thinmetalstrip/woodenbeamBéziercurves：PierreBézier:Renaultcarcompanyin19604.3IntroductiontoSplinesBéziercurveisaspline,itwasextendedtoB-spline,nonrationalB-splines,andtoNURBS.NURBScurve/surfaceisapowerfulcurve/surfacetoconstructirregularcurves/surfaces,anditiswidelyusedin3DCADsystem.4.3IntroductiontoSplines4.3IntroductiontoSplinesBeautyofcurves/surfaces:RomanPantheon4.3IntroductiontoSplines126AD43.3metres4.3IntroductiontoSplinesSt.Peter'sBasilicabeganin1506andwascompletedin16264.3IntroductiontoSplinesinternaldiameter:41.47metresFigure:Acubic(degree3)BeziercurveBéziercurvemimicstheshapeofitscontrolpolygon.MathematicsbehindBéziercurves

areinvestigatedinthis

section

sothattheycouldbemodeledmathematicallyonthecomputer.4.4BezierCurves/SurfacesAnnth-degreeBeziercurveisdefinedby:Thebasisfunctions,{Bi,n(u)},aretheclassicalnth-degreeBernsteinpolynomialsgivenby:{Pi}：controlpoints4.4BezierCurves/SurfacesExamples:n=1，Afirst-degreeBeziercurveE.g1:4.4BezierCurves/SurfacesC(u)=(1-u)P0+uP1

Thisisastraightlinesegmentfromp0top1B0,1(u)=1-u;B1,1(u)=un=2,C(u)=(l-u)2Po+2u(l-u)P1

+u2P2

Asecond-degreeBeziercurve.E.g2:ThisisaparabolicarcfromPotoP2

Noticethatthepolygonformedby{Po,Pl,P2},calledthecontrolpolygon,approximatestheshapeofthecurverathernicely4.4BezierCurves/Surfacesn=3,C(u)=(l-u)3Po+3u(l-u)2P1

+3u2(1-u)P2+u3P3

CubicBeziercurvesE.g3:Noticethatthecontrolpolygonapproximatestheshapeofthecurve.4.4BezierCurves/SurfacesConstructaBeziercurve:FromE.g2(n=2),wehaveC(u)=(l-u)2Po+2u(l-u)P1

+u2P2

then

Thus,C(u)isobtainedasthelinearinterpolationoftwofirst-degreeBéziercurves;inparticular,anypointonC(u)isobtainedbythreelinearinterpolations.4.4BezierCurves/Surfaces4.4BezierCurves/SurfacesAQuadratic(degree2)BeziercurveC(u0)=

ConstructhighdegreeBeziercurvebyrepeatedlinearinterpolationatt.Acubic(degree3)BeziercurveAquartic(degree4)Beziercurve

SubdivisionofBeziercurves4.4BezierCurves/SurfacesBezierpatch:Thepatchisconstructedfroman(n+1)×(m+1)arrayofcontrolpoints{Pi,j:0≤i≤n,0≤j≤m.},Andtheresultingsurface,whichisnowparameterizedbytwovariables,isgivenbytheequation:Bezierpatch4.4BezierCurves/Surfaces

PropertiesofBeziercurves/surfaces(1)Endpointinterpolation(2)Smoothnessgetasmoothcurveofthegivendegree.noticethedirectionsofp1-p0andpn-pn-1.howtomaketwobeziercurvesjoinsmoothly?(3)Convexhull(4)Variationdiminishingnostraightline/planeintersectsacurvemoretimesthanitintersectsthecurve’scontrolpolygonC(0）=P0,C(1)=P14.4BezierCurves/SurfacesForcurve:forall0≤u≤1Forsurface:forall0≤u≤1and0≤v≤1(5)Partitionofunity4.4BezierCurves/Surfaces(4)AffineinvarianceCurvesconsistingofjustonepolynomialorrationalsegmentareofteninadequate.Theirshortcomingsare:•ahighdegreeisrequiredtoaccuratelyfitsomecomplexshapes;•single-segmentcurves(surfaces)arenotwell-suitedtointeractiveshapedesign.•MostshapesaretoocomplicatedtodefineusingasingleBeziercurve.4.5B-splineThesolutionistousecurveswhicharepiecewisepolynomial.Figure：acurve,C(u),consistingofm(=3)nthdegreepolynomialsegmentsFor

example,apiecewisecollectionofBeziercurves,connectedendtoend,canbecalledasplinecurve.4.5B-splineFigure:AB-splinecurvewith8controlpointsAB-splineisageneralizationoftheBeizercurve.Itisasequenceofcurvesegmentsthatareconnectedtogethertoformasinglecontinuouscurve.HowtodefinetheB-splinebasisfunctions?Manyways4.5B-splineB-splinecurveisdefinedby：Where：{Pi}：controlpoints

kistheorderofthebasisfunctions(k=degree+1)

4.5B-splineAKnotvectorisalistofparametervalues,orknots,thatspecifytheparameterintervalsfortheindividualBeziercurvesthatmakeupaB-spline.For

example.

If

a

cubic

B-spline

is

comprised

of

four

Bezier

curves

with

parameter

intervals

[1,2],

[2,4],

[4,5],

and

[5,8],

the

knot

vector

would

be

:[u0,

u1,1,

2,

4,

5,

8,

u7,

u8]Theseextraknotscontroltheendconditionsofthecurve4.5B-splineTheKnotvectorisasequenceofparametervaluesthatdetermineswhereandhowthecontrolpointsaffectthecurve.m+1=（n+1)+km+1:#ofknotsn+1:#ofcontrolpointsK:orderofcurvem=n+k4.5B-splineKnotvectorscanbeclassifiedas:Periodic/Uniform,NonPeriodic/NonUniformPeriodic:B-splinebasisfunctionsarealltranslatesofeachother.4.5B-spline

NonPeriodic:krepeatedvaluesattheends(k=order)

NonUniform

knotspans：Theknotvectordividestheparametricspaceintheintervalsusuallyreferredtoas

knotspans.[0.0,1.0,2.0,3.0,3.0,5.0,6.5,7.0]4.5B-spline

A

(k=p+1)B-splinecurveisdefinedby:Normalizationofu:0≤u≤1The{Ni,k(u)}arethek-1degree(kistheorder)B-splinebasisfunctionsdefinedonthenonperiodic(andnonuniform)knotvector.Eachknotspanup+1≤u≤u(p+1)+1ismappedontoapolynomialcurvebetweentwosuccessivejointsC(up+1）andC(up+2）4.5B-splineThe

ith

B-splinebasisfunctionofpdegree(order=p+1),denotedbyNi,k(u),isdefinedas

Cox-deBoorrecursionformulaP=0P﹥0•Ifthedegreeiszero,allbasisfunctionsareall

stepfunctions.

Thedefinitionindicates:

Forexample,ifwehavefourknotsu0=0,u1=1,u2=2andu3=3,knotspansare[0,1),[1,2),[2,3)andthebasisfunctionsofdegree0areN0,0(u)=1on[0,1)and0elsewhere,N1,0(u)=1on[1,2)and0elsewhere,andN2,0(u)=1on[2,3)and0elsewhere.Thisisshownbelow:4.5B-spline•forp>0,Ni,p(u)isalinearcombinationoftwo(p-1)degreebasisfunctions;4.5B-splineHowtocomputeNi,k(u)forpgreaterthan0?P﹥0Usingthetriangularcomputationscheme4.5B-splineAllknotspansAllbasisfunctionswithdegree0Allbasisfunctionswithdegree1Allbasisfunctionswithdegree2Allbasisfunctionswithdegree3Allbasisfunctionswithdegree4Allbasisfunctionswithdegree5Tocompute

Ni,1(u),

Ni,0(u)and

Ni+1,0(u)arerequired,similarly,Tocompute

Ni,2(u),

Ni,1(u)and

Ni+1,1(u)arerequired,andsoon.

4.5B-splineForexample:Forknotvector:U

={0,1,2,3}Let’scomputeN0,1(u),

N1,1(u)

andN0,2(u)

Sincei=0andp=1,fromthedefinitionwehaveSinceu0=0,u1andu2=2,theabovebecomesSinceN0,0(u)isnon-zeroon[0,1),ifuisin[0,1),onlyN0,0(u)contributestoN0,1(u),theabovebecomesN0,1(u)=uN0,0(u)=u4.5B-splineIfuisin[1,2),onlyN1,0(u)contributestoN0,1(u),theabovebecomesN0,1(u)=（2-u）Similarly,wegetN1,1(u)=（u-1）ifuisin[1,2)Inthefigure,theblackandredlinesareN0,1(u)andN1,1(u),respectively.N1,1(u)=（3-u）ifuisin[2,3)4.5B-splineOnceN0,1(u)andN1,1(u)areavailable,wecancomputeN0,2(u).Plugginginthevaluesoftheknots:NotethatN0,1(u)isnon-zeroon[0,1)and[1,2)andN1,1(u)isnon-zeroon[1,2)and[2,3).Therefore,wehavethreecasestoconsider:uisin[0,1):4.5B-splineInthiscase,onlyN0,1(u)contributestothevalueofN0,2(u).SinceN0,1(u)isu,wehaveuisin[1,2):Inthiscase,bothN0,1(u)andN1,1(u)contributetoN0,2(u).SinceN0,1(u)=2-uandN1,1(u)=u-1on[1,2),wehaveuisin[2,3):Inthiscase,onlyN1,1(u)contributestoN0,2(u).SinceN1,1(u)=3-uon[2,3),wehave4.5B-splineIfwedrawthecurvesegmentofeachoftheabovethreecases,weshallseethattwoadjacentcurvesegmentsarejoinedtogethertoformacurveattheknots.

thecurvesegmentsofthefirstandsecondcasesjointogetheratu=1,whilethecurvesegmentsofthesecondandthirdcasesjoinatu=2.Notethatthecompositecurveshownhereissmooth.4.5B-splineHowtodeterminethenon-zerodomainofabasisfunctionNi,p(u),?Wecantracebackusingthetriangularcomputationschemeuntilitreachesthefirstcolumn.Conclusion:BasisfunctionNi,p(u)isnon-zeroon[ui,ui+p+1).Or,equivalently,Ni,p(u)isnon-zeroonp+1knotspans[ui,ui+1),[ui+1,ui+2),...,[ui+p,ui+p+1).4.5B-splineN1,3(u)isnon-zeroon[u1,u2),[u2,u3),[u3,u4)and[u4,u5).Or,equivalently,itisnon-zeroon[u1,u5).Oppositedirection:Givenaknotspan[ui

,ui+1),whichbasisfunctionswillusethisspaninitscomputation?4.5B-splineForexample,tofindalldegree3basisfunctionsthatarenon-zeroon[u4,

u5).drawatriangle(greenone),allfunctionsontheverticaledgesarewhatwewant.Inthiscase,theyare

N1,3(u),N2,3(u),

N3,3(u),N4,3(u).AB-splinesurfaceisobtainedbytakingabidirectionalnetofcontrolpoints,twoknotvectors,andtheproductsoftheunivariateB-splinefunctions.withUhasr+1knots,andVhass+1knots.r=n+p+1ands=m+q+1

4.5B-splineProperitiesofB-splinecurves/surfaces(1)Piecewisepolynomialcurve:4.5B-spline(2)Smoothness(3)Strongconvexhull(4)Variationdiminishing(5)Affineinvariance(6)PartitionofunityForcurve:forall0≤u≤1Figure:AB-splinecurvewithdifferentcontinuityTransformscurvetransformscontrolpolygonLieswithinunionofconvexhullsofk4.5B-splineForsurface:forall0≤u≤1and0≤v≤14.5B-spline(7)LocalmodificationschemeMovingPichangesaB-splinecurveonlyintheinterval[ti,ti+k),becauseNi,k(t)=0fort

∉[ti,ti+k);similarly,movingPi,jonlyaffectsaB-splinesurfaceintherectangle[ui,ui+p+1)×[vj,vj+q+1),becauseNi,p(u)Nj,q(v)iszeroif(u,v)isoutsideofthisrectangle.MovingthecontrolpointB7onlychangesthecurvenearthatpoint.LocalizedControlExample4.5B-spline4.6IntuitiveUnderstandingofB-splinesBasisfunctionsNi,k(t)Ni,k(t),determineshowstronglycontrolpointsBiinfluencesthecurveattimet,iscalledthebasisfunctionforthatcontrolpoint.EachcontrolpointhasonebasisfunctionFigureBasisfunctionforacontrolpointAtt=3,controlpointBihasitsgreatesteffect(about95%)Ni,k(t)=Ni+1,k(t+Δt)=Ni+2,k(t+2Δt)Figure

n=k=3periodic/uniformbasisfunctionPeriodic:B-splinebasisfunctionsarealltranslatesofeachother.

4.6IntuitiveUnderstandingofB-splinesNoticethatinfluenceofbasisfunctionislimitedtokintervals4.6IntuitiveUnderstandingofB-splines4.6IntuitiveUnderstandingofB-splinesFigure:

UniformbasisfunctionsforasetofcontrolpointsU={0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0}Noticethatallofthebasisfunctionsinthefigurehaveexactlythesameshapeandcoverequalintervalsofthetime–Periodic/UniformB-splinesEx1:

n=4,k=3periodic/uniformbasisfunctionFigure:

curvewithuniformknotvector4.6IntuitiveUnderstandingofB-splinesSofar,noticethat:(1)Endpointsofthecurvejust“floatinginspace”;thatis,thecurve’sendpointsdon’tcoincidewithanycontrolpoint.(2)Curvesarequitesmooth,whilewesometimesneedtocreateacurvewithakinkorcornerHowtoaccomplishtheabovegoals?Givingseveralconsecutiveknotsthesamevalueoft!NonperiodicB-splinesNonPeriodic:krepeatedvaluesattheends(k=order)

4.6IntuitiveUnderstandingofB-splinesFigure:BasisfunctionsforacurvewithmultipleidenticalknotsatthebeginningU={0.0,0.0,0.0,3.0,4.0,5.0,6.0,7.0}Figure:curvewithmultipleidenticalknotsatthebeginningNoticethatN0,3(t)(theoneforcontrolpointB0)hastotalcontroloverthecurveatt.Thusthecurveatt

=

0coincideswiththefirstcontrolpoint.NonPeriodic:Thecurveinterpolatesthefirstcontrolpoint4.6IntuitiveUnderstandingofB-splinesInpractice,wehopesomecontrolpointsaffectalargerregionofthecurveandothersasmallerregion,sothatmorecomplexcurvescanbecreated.Howtoaccomplishtheabovegoals?varythewidthoftheintervals!non-uniformB-splinesEx2:

ChangingtheknotvectorinEx1toU={0.0,1.0,2.0,3.75,4.0,4.25,6.0,7.0}Ex1Ex24.6IntuitiveUnderstandingofB-splines4.6IntuitiveUnderstandingofB-splinesKnotsandKinks?Sofar,curvesdisplayedarequitesmoothHowever,wesometimesneedtocreateacurvewithakink,cornerorcreaseExamplesofcreasesHowtocreateacurvewithakinkorcorner?Bunchupsomeknotsinthemiddleoftheknotvector,forexample{0.0,1.0,2.0,3.0,3.0,5.0,6.0,7.0}4.6IntuitiveUnderstandingofB-splinesBunchupsomeknotsinthemiddleoftheknotvector,forexample{0.0,1.0,2.0,3.0,3.0,5.0,6.0,7.0}Figure：BasisfunctionsforacurvewithmultipleidenticalknotsinthemiddleFigure：curvewithmultipleidenticalknotsinthemiddleAtt=3,allthebasisfunctionsexceptN2,3(t)have0value—socontrolpointB2istheonlyonetoaffectthecurveatthatinstant,andthuscurvecoincideswiththatcontrolpoint---creatingakink4.6IntuitiveUnderstandingofB-splinesIftwoknotscoincide,thecontinuityatthatjointgoesdownbyonedegree;ifthreecoincide,thecontinuitygoesdownanotherdegree;andsoon.Thismeansyoucanputakinkinthecurveataparticularpointbyaddingknotstotheknotvectoratthatpoint.Non-UniformB-splinesNURBcurvesNon-UniformRationalB-splines:NURBS

Whatdoes“Rational”mean?4.6IntuitiveUnderstandingofB-splinesHowtomakeindividualcontrolpointhavedifferentinfluenceontheshapeofthecurve?GiveaweightwforeachcontrolpointIncreasingtheweightofanindividualcontrolpointgivesitmoreinfluenceandhastheeffectof“pulling”thecurvetowardthatpoint.Figure:IncreasingtheweightofacontrolpointCurveswithaweightforeachcontrolpoint,arecalledrationalcurves.Notice:Iftheweightsaresetto1,theNURBSbecomesaregularB-spline4.7NURBS2.1RewritingC(u)intheform:4.7NURBS4.7NURBSTheNURBSsurfaceEq.canbewrittenas:ConclusionsANURBScurve/surfaceisdefinedbyfourthings:degree,controlpoints,knots.TwoormoreNURBScurvescanbecombinedinmost3DapplicationstocreateaNURBSsurface4.7NURBSNURBSdifferfromB-splinesmainlyintwoaspects:nonuniform,andrationalProperitiesofNURBScurves/surfacesPiecewiserationalpolynomialcurve.Endpointinterpolation:NURBScurvesinterpolatethefirstandthelastcontrolpoints:P0=C(0)andPn=C(1);NURBSsurfacesinterpolatethefourcornercontrolpoints.4.7NURBS

Strongconvexhull

Variationdiminishing

Projectiveinvariance

Partitionofunity

Localmodificationscheme4.7NURBSThedegreeofthecurvedeterminesthesmoothnessofthejoinsbetweenspans.SmoothjoinsDegree1(linear)curvesgivepositionalcontinuityatthejoin,degree2(quadratic)curvesgivetangentcontinuityanddegree3(cubic)curvesgivecurvaturecontinuity.4.7NURBSFigure:continuityofacurve5.Continuity:Figure:continuityofacurvegeometriccontinuity4.8ContinuityofCurves/SurfacesZebraFigure:continuityofasurface4.8ContinuityofCurves/SurfacesCurvature(G2)Positional(G0)Tangent(G1)4.8ContinuityofCurves/SurfacesC0,C1,C2continuity:parametriccontinuity

G0,G1,andG2continuityisindependentoftheparameterizationofthecurveorsurface.C0,C1,andC2continuityisdependentontheparameterizationofthecurveorsurface.Ingeneral,CcontinuityismorestringentthanGcontinuity.Forexample,C2continuityalwaysimpliesG2continuity,andC1continuityalwaysimpliesG1continuity,butnotviceversa.G1-2andC1-2aresimilarbutnotquitethesamething.4.8ContinuityofCurves/Surfaces

Inthepastseveralyearstrianglemesheshavebecomeincreasinglypopularandarenowadaysintensivelyusedinmanydifferentareasofcomputergraphicsandgeometryprocessing.Subdivisionoftrianglemeshes4.9SurfacesBasedonTriangleMeshesFractal(geometry)

In

mathematics

a

fractal

isan

abstractobject

usedtodescribeandsimulatenaturallyoccurringobjects.Artificiallycreatedfractalscommonlyexhibitsimilarpatternsatincreasinglysmallscales.

Itisalsoknownas

expandingsymmetry

or

evolvingsymmetry.Ifthereplicationisexactlythesameateveryscale,itiscalleda

self-similar

pattern.4.9SurfacesBasedonTriangleMeshesFractal(geometry)Fractalsarenotlimitedtogeometricpatterns,butcanalsodescribeprocessesintime

Fractalpatternswithvariousdegreesofself-similarityhavebeenrenderedorstudiedinimages,structuresandsoundsandfoundinnaturetechnology

art

and

law.Fractalsareofparticularrelevanceinthefieldof

chaostheory,sincethegraphsofmostchaoticprocessesarefractal.4.9SurfacesBasedonTriangleMeshesExamples:

There's

only

one

corner

of

the

universe

you

can

be

sure

of

improving,

and

that'syour

own

self.THEENDPart5:3DPrintingCAD/CAM:PrinciplesandApplications(CAD/CAM原理与应用)CollegeofMechanicalandElectricalEngineering,HohaiUniversity（河海大学机电工程学院）Associateprof.康兰

Historyof3DPrinting3DPrintingProcess3DPrintingMaterialsContents3DPrintingBenefitsandChallengesWhatis3DPrinting?PrinciplesBehind3DPrintingAnOverviewof3DPrintingIndustryContentsApplicationsandtheFutureof3DPrintingAcquiringa3DModelTheEmergingApplicationsof3DPrinting4Dprinting

Will3DPrintingReplaceConventionalManufacturing?

CaseStudiesHistoryof3DPrintingThetechnologyforprintingphysical3DobjectsfromdigitaldatawasfirstdevelopedbyCharlesHullin1984.Historyof3DPrintingHullnamedthetechniqueasStereolithography

andobtainedapatentforthetechniquein1986.（Stereolithography）Hullwentontoco-found3DSystemsCorporation(oneofthelargestandmostprolificorganizationsoperatinginthe3Dprintingsectortoday.)Bytheendof1980s,othersimilartechnologiessuchasFusedDepositionModeling(FDM)andSelectiveLaserSintering(SLS)wereintroduced.Historyof3DPrintingIn2005,ZCorp.launchedabreakthroughproduct,namedSpectrumZ510,whichwasthefirsthighdefinitioncolor3DPrinterinthemarket.Anotherbreakthroughin3DPrintingoccurredin2008withtheinitiationofanopensourceproject,namedReprap,whichwasaimedatdevelopingaself-replicating3Dprinter.Inthepastyears,researchersandscientistsweretryingtoexplorenewapplicableareasof3Dprinting,especiallyinmedical

industryandaerospaceindustry.Studyingnewmaterialssuitablefor3Dprinting.Historyof3DPrintingBioprinter,4Dprinting.“3Dprintedcar”,“3Dprintedkidney”,“3Dprintedmetalparts”,……AnOverviewof3DPrintingIndustry1.

InfluentialCompaniesin3DPrintingIn2014,Appinionsliststhe“TenMostInfluentialCom