Presenters:ZhaoHui,YangGuoxingTianjinLightIndustryVocationalTechnicalCollege1.UnderstandingSlopeandTaperImportDirectoryConceptofSlopeDrawingandLabelingSlope12UnderstandingSlopeandTaperConceptofTaper3DrawingandLabelingTaper4ConceptofSlopeS=tanβ=(H-h)/LUsually,theantecedentoftheratioisreducedto1,andtheslopeisexpressedintheformofasimplefraction1:n.MethodofDrawingSlope1:5BCCHENWPBCCHENWPSlopeExampleDrawingSteps①Drawtheknownstraightsectionsbasedonthegivendimensionsinthediagram.②FrompointA,constructaright-angledtriangleata1:12slope,thendeterminethehypotenuseAC.③FromthegivenpointD,drawalineparalleltoAC.④Darkenandthickenthewedgekeyoutline,thenmarktheslopesymbol.Thebottomlineoftheslopesymbolshouldbeparalleltothereferencesurface(line),andthedirectionofthesymbol'stipshouldbeconsistentwiththeinclinationdirectionoftheslope.DrawingoftheSlopeSymbol.ConceptofTaperC=(D-d)／L=2tan(α／2)Liketherepresentationofslope,theantecedentofthetaperratioisusuallyreducedto1andwrittenintheformof1:n.TaperExampleLatheCenterToolShankMethodofDrawingTaperTakehalfofthetaperintheverticaldirection1:4TaperExampleDrawthegraphicwitha1:5taperasshowninthetaperaxisDrawingSteps①Accordingtothedimensionsinthedrawing,drawtheknownstraightlinesection.②ArbitrarilydeterminethebaseABoftheisoscelestriangleas1unitlengthandtheheightas5unitlengths,anddrawtheisoscelestriangleABC.③FromtheknownpointsDandE,drawlinesparalleltoACandBCrespectively.④Darkenandthickenthegraphicandlabelthetapersymbol.SummaryConceptualDifferenceBetweenSlopeandTaperDrawingMethodsofSlopeandTaper12UnderstandingSlopeandTaperFutureCareerApplicationWhyarebulletsmadetapered?Inourfuturework,weshouldcombinepracticalworkwithmorethinkingandreflection.Onlybythinkingdiligentlycanwecontinuetoprogress.Whatisthetaperofthetoolholderinamachiningcenter?PracticeTaskPresenters:ZhaoHui,YangGuoxingTianjinLightIndustryVocationalTechnicalCollege2.ArcConnectionIntroductionIntroductionContentsConceptofarcconnectionTypesofarcconnection12ArcconnectionDrawingmethodofarcconnection3ConceptofarcconnectionArcconnectionisadraftingmethodthatemploysatangentarctosmoothlyjointwoadjacentsegments(linesorarcs)withoutabrupttransitions.Essenceofarcconnection:thearcistangenttothestraightline,andthearcistangenttothearc.SmoothTypesofarcconnectionSelectanypointonL1orL2ascenterO;anarcdrawnwithradiusRwillbetangenttolineL.

ConclusionArctangenttoaknownstraightline01TypesofarcconnectionTakeanypointontheconcentriccircleasthecenterO,anddrawanarcwithradiusR.Thearcistangenttotheknownarc.KConclusionArctangenttoaknownarcexternally02TypesofarcconnectionTakeanypointontheconcentriccircleasthecenterO,anddrawanarcwithradiusR.Thearcistangenttotheknownarc.ConclusionKArctangenttoaknownarcinternally03DrawingmethodofarcconnectionKnownradiusFixedcenterFindtangentpointConnectarcDrawingmethodofarcconnectionoRRBK2K1ORRRSpecialmethodofarcconnectingtwostraightlinesatarightangleMethodofarcconnectingtwostraightlinesatanyangleArctangenttoaknownstraightline01DrawingmethodofarcconnectionRa=R1+RRb=R2+RRMethod:Thecenterliesattheintersectionofarcswithradii(R+R1)and(R+R2).Arctangenttotwoknownarcsexternally02DrawingmethodofarcconnectionRa=R-R1Rb=R-R2Method:findthecenterbythedifferenceofthetwointernaltangentradiiArctangenttotwoknownarcsinternally03DrawingmethodofarcconnectionRT1T2OR+R1R+R2R1O1O2R2RR2O2R1O1R-R1R-R2T2T1RORSummaryConceptandtypesofarcconnectionDrawingmethodofarcconnection12ArcconnectionPracticeTask主講人：趙慧、楊國星TianjinLightIndustryVocationalTechnicalCollege3.DrawingtheThree-ViewProjectionofaStraightLinePartIIn-ClassReviewPARTIIn-ClassReviewI.Three-ViewProjectionofaStraightLineDefinition：Astraightlineisdeterminedbytwopoints.Byconnectingthecorrespondingprojectionsofthesetwopointswithastraightlineineachview,weobtainthecorrespondingprojectionofthestraightline.aa

a

b

b

b●●●●●●In-ClassReviewII.ProjectionCharacteristicsofaStraightLineonaSingleProjectionPlaneAB●●●●abStraightlineperpendiculartotheprojectionplane:theprojectioncoincidesintoasinglepoint(AccumulationProperty)Straightlineparalleltotheprojectionplane:theprojectionreflectsthetruelengthofthesegmentab=AB(True-Length

Property)Straightlineinclinedtotheprojectionplane:theprojectionisshorterthanthespatialsegmentab=ABcosα(Similarity

Property).●●AB●●abαAMB●a≡b≡m●●●In-ClassReviewIII.ProjectioncharacteristicsofastraightlineinthesystemofthreeprojectionplanesProjection-plane-parallel

linesProjection-plane-perpendicular

lines

frontalline—paralleltotheV-planeprofileline—paralleltotheW-planehorizontalline—paralleltotheH-planefrontalperpendicularline—perpendiculartotheV-planeprofileperpendicularline—perpendiculartotheW-planeVerticalline—perpendiculartotheH-planeGeneral‐positionstraightlinecollectivelyreferredtoasspecial-positionlinesStraightlineinclinedtoallthreeprojectionplanesPerpendiculartooneprojectionplanewhileparalleltotheothertwoprojectionplanes.Parallel

tooneprojectionplaneandInclinedtotheothertwoprojectionplanes.PartIIProjection-planeperpendicular

linesPARTIIProjection-planeperpendicular

linesProjection-planeperpendicular

lines(⊥,∥,∥)1.Verticalline（⊥totheH-plane，∥totheV-plane，∥totheW-plane）b'b"a"BA0WHVXZb(a)a

b

b(a)a''b

ZXYWYHa'Projection-planeperpendicular

linesProjection-planeperpendicular

lines(⊥,∥,∥)2.

Frontalperpendicularline（⊥totheV-plane，∥

totheH-plane，∥

totheW-plane）a'(b')b"a"BA0WHVXZYabXZYH0abb"a"a'(b')Projection-planeperpendicular

linesProjection-planeperpendicular

lines(⊥,∥,∥)3.

Profileperpendicularline（⊥totheW-plane，∥totheV-plane，∥totheH-plane）a"(b")Wa'BA0HVXZb'abYWZYH0aba"(b")a'b'XYProjection-planeperpendicular

linesProjection-planeperpendicular

lines(⊥,∥,∥)Projectioncharacteristics:(1)Onitsperpendicularprojectionplane,

theprojectionconvergesintoapoint.(2)Theprojectionsontheothertwoplanes,

reflecttheactuallengthofthesegmentandareperpendiculartothecorrespondingprojectionaxes.a

b

a(b)a''b

ZXYWYHb'(a)'baa

b

ZXYWYHa'b'aba''(b

)ZXYWYHIdentificationmethod:"Onepoint,two

lines"–theprojectionthatisapointononeprojectionplaneindicatesitisperpendiculartothatprojectionplane.ProfileperpendicularlineFrontalperpendicularlineVerticallinePartIIIProjection-planeParallelLinesPARTIIIProjection-planeParallelLines（1）HorizontalLine（∥Hplane，∠Vplane，∠Wplane）

Projection-planeParallelLines(∥,∠,∠)）truelengthba''aa

b'b''ZXYHYWβγXZYOaa

b

a

bb

ABβγγβProjection-planeParallelLines（2）FrontalParallelLine（∥Vplane，∠H

plane，∠W

plane）Projection-planeParallelLines(∥,∠,∠)truelengthαγααXb"b'ba'a"aoZHWVABXaboa'b'ZYHb"a"YWYγγααProjection-planeParallelLines（3）ProfileLine（∥W

plane，∠V

plane，∠H

plane）Projection-planeParallelLines(∥,∠,∠)）truelengthXaba'b'ZYHb"a"YWob"b'ba'a"aZHWVABXYoαααβββProjection-planeParallelLinesProjection-planeParallelLines(∥,∠,∠)Projectioncharacteristics:Ontheprojectionplanetowhichthelineisparallel,itsprojectionshowsthesegment’struelengthandthetruemagnitudeoftheanglesbetweenthelineandtheothertwoprojectionplanes.

Ontheothertwoprojectionplanes,theprojectionsareparalleltotherespectiveprojectionaxes.Identificationmethod:“oneinclined,two

parallel”—theprojectionthatappearsinclinedonagivenprojectionplaneindicatesthatthelineisparalleltothatprojectionplane.b

a

aba

b

b

aa

b

ba

γAnglewiththeH-plane:α;anglewiththeV-plane:β;anglewiththeW-plane:γβγααβba

aa

b

b

HorizontallineFrontalparallellineProfileparallellinetruelengthtruelengthtruelengthProjection-planeParallelLinesProjection-planeParallelLines(∥,∠,∠)ProjectionCharacteristics:(1)Ontheprojectionplanetowhichthelineisperpendicular,projectionconvergestoapoint(i.e.,exhibitsconvergence).(2)Ontheothertwoprojectionplanes,theprojectionsrepresentthetruelengthofthelinesegmentandareperpendiculartotheirrespectiveprojectionaxes.a

b

a(b)a''b

ZXYWYHb'(a)'baa

b

ZXYWYHa'b'aba''(b

)ZXYWYHIdentificationMethod:"Onepoint,two

lines"—Theplaneonwhichtheprojectionappearsasapointindicatesthatthelineisperpendiculartothatprojectionplane.ProfileperpendicularLineFrontalperpendicularlineVerticalline?PartIVLineinGeneralPositionPARTIVLineinGeneralPositionLineinGeneralPosition(∠,∠,∠)OXZYABbb

a

b

aa

ZXa

b''aOYHYWa

bb

Allprojectionsonthethreeprincipalplanesareinclinedstraightlines,andallofthemareshorterthanthetruelength.ProjectionCharacteristics:IdentificationMethod:"TripleInclination".PartVKnowledge

ReinforcementPARTVKnowledge

ReinforcementDeterminethepositionaltypeofeachofthefollowinglines:GeneralPositionLineFrontalParallelLineVerticalLine?true

lengthb

a

aba

b

ZXYWYHYHYWXZZXYWYHa

b

a(b)a

b

a

b

abb

a

SummaryofThisLecture0102ClassificationoftheProjectionsofaStraightLineProjectionCharacteristicsofaStraightLineintheThree-PlaneProjectionSystem03HowtoDeterminetheSpatialOrientationofaStraightLineThoughtQuestionsLineACisa(n)

line.LineSBisa(n)

line.LineSAisa(n)

line.DeterminingtheRelativePositionsofEachEdgeofaTriangularPyramidtotheProjectionPlanesSpeakers:ZhaoHui,YangGuoxingTianjinLightIndustryVocationalTechnicalCollege4.DrawingtheThree-ViewProjectionofaSphereSphere

k

k

kCircle’sRadius?Thecirculargeneratrixisrotatedaboutitsdiameter,whichservesastheaxisofrotation.Thethreeviewsarethreecircleswithadiameterequaltothatofthesphere,whicharetheprojectionsofthesphere’scontourlinesinthreedirections.FormationoftheSphericalSurfaceProjectionoftheSphere(ThreeViews)SelectingPointsontheSphere’sSurfaceAuxiliaryCircleMethodIntersectionLineofSphereWhenaplaneintersectsasphere,theintersectionlineisalwaysacircle.Dependingontherelativepositionbetweenthecuttingplaneandtheprojectionplane,

theprojectionoftheintersectionlinemaybeacircle,anellipse,orastraightline.IntersectionLineofSphereForthetheprojectionsoftheintersectionlinesgeneratedbyahorizontalplanecuttingthesphere:Inthetopview,theprojectionappearsasapartialcirculararc;Intheleftview,itconvergesintoastraightline.Forthetheprojectionsoftheintersectionlinesgeneratedbytwosideplanescuttingthesphere:Intheleftview,theprojectionappearsaspartialcirculararcs;Inthetopview,itconvergesintoastraightline.ExampleDeterminethetopviewandleftviewofahemisphereafterbeingcut.ExampleDeterminetheplanviewandleftviewofahemisphereafterbeingcut.Exercise:Speakers:ZhaoHui,YangGuoxingTianjinLightIndustryVocationalTechnicalCollege5.DrawingtheThree-ViewProjectionofaCylinderCylinder

CommonBasicGeometricSolidsPlaneSolidsCurvedSolidsCylinderAnylineonthecylindricalsurfacethatisparalleltotheaxisiscalled

aprofileline.Acylinderisasolidboundedby

acylindricalsurfaceandtwocircularbases.AcylindricalsurfacecanberegardedasalineAA1rotatingaboutalineOO1thatisparalleltoit.ThelineAA1iscalledthegeneratrix.A1AOO1FormationoftheCylindricalSurface:CylinderStepsforDrawingtheThree-ViewProjectionofaCylinder

A1AOO1SelectingPointsontheCylindricalSurfaceA1AOO1Utilizingtheaccumulationofprojections.Giventheprojections(1′,2′,3′,4)ofpointsonthecylindricalsurface,determinetheirprojectionsintheothertwoviews.Cylinder

3

3

1′

1

4"

(2

)

2"

2

4

4

1"KnowledgeExplanationIntersectionLine:thelineformedbytheintersectionofaplanewiththesurfaceofasolid.Theplaneiscalledthecuttingplane,andtheresultingsolidisknownasasectionedsolid.

Duetothedifferentrelativepositionsbetweenthecuttingplaneandthecylinder’saxis,theintersectionlinecantakeonthreedifferentshapes:CircleEllipseRectanglePerpendicularInclinedParallelCylinderCutbyaPlaneKnowledgeExplanation(1)SpatialandProjectionAnalysisAnalyzetheshapeofthesolidofrevolutionandtherelativepositionbetweenthecuttingplaneandthesolid’saxis.Analyzetherelativepositionbetweenthecuttingplaneandtheprojectionplanes(e.g.,accumulation,similarity).Identifytheknownprojectionoftheintersectionlineandpredictitsunknownprojection.(2)DrawtheProjectionoftheIntersectionLineWhentheintersectionline’sprojectionisanon-circularcurve,followthesesteps:Smoothlyconnectallpointsandevaluatethe

visibility

oftheintersectionline.Locatethespecialpoints(pointsontheoutlineprofilelineandatextremepositions).Determinetheshapeoftheintersectionline.DeterminetheprojectioncharacteristicsoftheintersectionlineAddthegeneralpoints.(3)RefinetheOutline.StepstoDeterminetheIntersectionLine:ExampleAcylinderisobliquelycutbyaverticalplane;determineitsthree-viewprojection.Whatistheshapeofthesideprojectionoftheintersectionline?Whatistheknownprojectionoftheintersectionline?Whatistherelationshipbetweenthecuttingplaneandthecylinder’saxis?(1)Analysis:★Locatethespecialpoints.★Addthegeneralpoints.★Smoothlyconnectallpoints.(3)RefinetheOutline(2)DeterminetheIntersectionLine:●5●6●8●71●3●2●4●●1'●2'(4')●3'●1"●2"●3"●4"●5'(6')●7'(8')●5"●7"●8"●6"ResultsandtheSolidDiagram●1'●2'(4')●3'●1"●3"●4"●5'(6')●7'(8')●5"●8"●6"●5●6●8●71●3●2●4●●1●3●4●2Completethehorizontalandsideprojectionsofthecylinder’scutsection.ExerciseSpeakers:ZhaoHui,YangGuoxingTianjinLightIndustryVocationalTechnicalCollege6.DrawingtheThree-ViewProjectionofaConeConceptsofaConeandItsThree-ViewProjectionO1ONotetheprojectionoftheoutlineprofilelineandtheevaluationofthesurface’svisibility.SAAconeisasolidboundedby

aconicalsurfaceandacircularbase.ConesFormationoftheConicalSurfaceTheconicalsurfaceisformedbyrotatinglineSA(thegeneratrix)aboutitsintersectinglineOO1.ApexProfileLineThree-ViewProjectionoftheConeStepsforDrawingtheThree-ViewProjectionofaConeO1OSA

s

s

sacbda

c

b

(d

)d

b

a

(

c

)SelectingPointsontheConesO1OSAGiventheProjection1、2、3,ontheconicalsurface,determineitsprojectionsintheothertwoviews.

s

s

(2

)

sacbda

c

b

(d

)d

b

a

(

c

)

1

1

1

2

2

(3)

3

3

SpecialPositionPointsSelectingPointsontheConicalSurfaceHowcanyouconstructastraightlineontheconicalsurface?Giventheprojections

1、2ofapointontheconicalsurface,determineitsprojectionsintheothertwoviews.GeneralPointsAuxiliaryProfileLineMethod:AuxiliaryCircleMethod:Drawanauxiliaryprofilelinethroughtheapex.●SM

s

●

s

●

1

(2

)s●2

1(2

)●

1

m

mPlaneCuttingaConeWhenthecuttingplaneisinclinedrelativetotheaxisθ>αEllipseCircleWhenthecuttingplaneisperpendiculartotheaxisθ=90°HyperbolaWhenthecuttingplaneisparalleltotheaxisθ=0°ParabolaWhenthecuttingplaneisparalleltooneprofilelineθ=αIsoscelestriangleWhenthecuttingplanepassesthroughtheapexWhatisthespatialshapeoftheintersectionline?Whatistheknownprojectionoftheintersectionline?Whatistherelationshipbetweenthecuttingplaneandthecone’saxis?★LocatethespecialpointsHowtolocatetheendpointsoftheotheraxisoftheellipse(i.e.,theforemostandrearmostpoints):★Addtheintermediatepoints.★Smoothlyconnectallpoints3.RefinetheOutline1.Analysis2.DeterminingtheIntersectionLine1'2'3‘(4’)5'(6')1"2"3"4"127‘(8')9‘(10')7"8"??5"6"??9"10"??78??910??43??56??AConeCutbyaVerticalPlane–CompletingtheThree-ViewProjectionPlaneCuttingaConeThree-viewprojectionandsoliddiagramofaconecutbyaverticalplane.1'2'3‘(4’)5'(6')7‘(8')9‘(10')7"8"??5"6"??9"10"??78??910??43??56??212"1"ExercisePresenters:ZhaoHui,YangGuoxingTianjinLightIndustryVocationalTechnicalCollege7.DrawthethreeviewsofaregularprismCommonBasicGeometricBodiesBasicplanarsolidBasiccurvedsolidCaseIntroductionGeometricsolidPlanarsolidCurvedsolidThosewithallsurfacesbeingflatarecalledplanarsolidsThosewithcurvedsurfacesarecalledcurvedsolidsCuttingbodyofaregularhexagonalpyramidCuttingbodyofaregularquadrangularprismDrawingviewsofplanarsolidsDrawprojectionsofalledges(oredgefaces),andrepresentthemwiththicksolidlinesordashedlinesbasedontheirvisibility.Edge:Thelineofintersectionbetweenadjacentsurfacesonasolid.ProjectionofplanarsolidsandtheirintersectionlinesArightprismwithtopandbottomfacesasregularpolygonsiscalledaregularprism.Boththetopandbottomfaceslieonhorizontalplanes,withtheirhorizontalprojectionsexhibitingcongruence.Thefrontandbackfacesarefrontalplanes,andtheotherfourfacesareverticalplanes;theirhorizontalprojectionsallaccumulateintolines,coincidingwiththesidesofthehexagon.I.Prism(1)ThreeviewsofaprismStepsfordrawingthreeviewsofaprismProjectioncharacteristicsofaprism:Projectiononaplaneparalleltothecharacteristicfaceisapolygon;TheothertwoprojectionsareseveralrectanglesStepsfordrawingthreeviewsofaprism(2)TakingpointsonthesurfaceofaprismVisibilityrulesforpoints:Iftheprojectionoftheplanewherethepointislocatedisvisible,theprojectionofthepointisalsovisible;Iftheprojectionoftheplaneaccumulatesintoaline,theprojectionofthepointisalsovisible.

a

a

a

(b)

b

b

cCCC

DrawthethreeviewsofaregularprismTheintersectionlineformedonthesurfaceofasolidwhenaplanecutsthroughitiscalledtheintersectionline.Theplanethatcutsthroughthesolidiscalledthecuttingplane.Thesolidthathasbeencutbytheplaneiscalledthetruncatedsolid.Propertiesoftheintersectionline:Theintersectionlinerepresentsthesharedboundarybetweenthetruncatingplaneandthesolid'ssurface.Theintersectionlineisaclosedplanefigure.IntersectionlineofaplanesolidAplanarsolidundergoestruncation,theintersectionlineisaclosedplanepolygon.Theverticesoftheresultingpolygoncorrespondtotheintersectionpointsbetweenthecuttingplaneandtheedgesofthesolidobject.Thecuttinglineisobtainedbycomputingeithertheintersectionpointsofedgeswiththecuttingplaneortheintersectionlinesoffaceswiththecuttingplane.Eachsideofthesectionpolygonrepresentstheexactintersectionlocuswherethecuttingplaneintersectswithaspecificfaceofthethree-dimensionalsolid.Exampleanalysisa'aa"b'bb"c(c')c"d'(e')dee"d"Pv1FindtheintersectionpointsoftheedgeswithplaneP.2Connectthepoints—connectpointsonthesamesurface.Findtheintersectionlineafteraquadrangularprismiscut.Three-viewprojectionPracticeTaskCompletethetopandsideviewsofthehexagonalprismafterithasbeencut.1(3)2(4)1

(2

)2"●1"●3

(4

)主講人：趙慧、楊國星TianjinLightIndustryVocationalTechnicalCollege8.DrawingtheThree-ViewProjectionofaRegularPrismPyramid(1)TheThreeViewsofaPyramidAsolidformedbyabaseandseverallateralfaces.Thelateraledgesconvergeatasinglepointatafinitedistance—theapex.Tofacilitatetheanalysisoflinesandsurfacesintheviews,theprojectionsofeachvertexmaybelabeled.Pyramid

s

b

s

a

c

abc

a

(c

)b

s

yyStepsforDrawingtheThree-ViewProjectionofaPyramid:Giventhehorizontal

projection“a”ofpointAonthepyramid’ssurface,determineitsprojectionsontheothertwoviews.(2)SelectingPointsonthePyramid’sSurfacea(a′)(a″)PyramidSectionedTriangularPyramidThinkingFindthetopviewandtheleftviewofthequadrangularpyramidafterithasbeencut.3

2

1

(4

)1

?1?3

?2

?4

?3?2?4?主講人：趙慧、楊國星TianjinLightIndustryVocationalTechnicalCollege9.DrawaStandardIsometricDrawingPartIIntroductionPARTIIntroduction1810162582036Canyouunderstandthissetofthree-viewdrawingsandmentallyvisualizeitsthree-dimensionalshape?Introduction1810162582036StandardIsometric

DrawingAdvantagesItallowsyoutoclearlyexpressthethree-dimensionalshapeyouhaveconceptualizedbasedonthethreeviews;Itenablescomparisonwiththethree-viewdrawings,helpingyoutocorrectanyerrorsinyourconstructed3Dimage;Forslightlymorecomplexthree-viewdrawings,youcanobserve,imagine,andsketchtheisometricdrawingsimultaneously.PARTIIFormationofAxonometricProjectionPARTIIFormationofAxonometric

ProjectionVHYAnobjectprojectsontothehorizontalprojectionplane,formingthetopviewACBZXOAnobjectprojectsontothefrontalprojectionplane,formingthefrontviewFormationofAxonometricProjectionVHPYDrawingAxonometricProjectionsIntroducetheinclinedplaneaxonometricprojectionplaneACBZXOYZ1Y1X1C1B1A1ThefigureobtainedbyprojectingtheobjecttogetherwiththeCartesiancoordinatesystemontothisprojectionplaneusingparallelprojectionaxonometricprojectionThefigureobtainedwhentheprojectiondirectionisperpendiculartotheaxonometricprojectionplaneisometric

drawingSAxonometricProjectionPlaneAxonometricProjectionDirectionIsometric

DrawingO1FormationofAxonometric

ProjectionVHPYACBZXOYZ1Y1