XXXX秋季第9讲换面机械制图北京航空航天大学出版社

例1：求已知直线与平面的夹角的实际大小。Findthetrueanglebetweenthegivenlineandplanem'

n'

nm步骤：求MN与△ABC的交点K(求交)PiercingpointofMNand△ABCa'c'b'abc1'122'

d'

dPvk'

k••过M作MD△ABCDrawMD

△ABC求MD与△ABC的交点S(求交)PiercingpointofMDand△ABC求MKL(△MKL)的实形即

(分别求出MS、KS的实长)ThetruesizeofMKL(△MKL)is

MKS

s'

s先求得KS、MS

的实长FindthetruelengthofKSandMSMPNSKθ例1：求已知直线与平面的夹角的实际大小。

Findthetrueanglebetweenthegivenlineandplanem'

n'

nm步骤：a'c'b'abc1'122'

d'

de'

e••过M作MD△ABCDrawMD

△ABC

NDM

EMN求NMD(△MND)的实形TruesizeofNMD(△MND)NMD的余角即为ThecomplementaryangelofNMDis需求得MN、ND、MD的实长如画水平线NE，只需求MN、ME的实长MPNSKθDE不求交？Withoutfindingtheintersection?问题：如何求异面直线的公垂线？Problem：Howtoconstructcommonperpendiculartoskewlines?如果其中一条直线是投射线？Whatifoneoftheskewlinesperpendiculartoprojectionplane?第四章投影变换Projectiontransformation一、问题的提出Problems如何求一般位置平面图形的真实大小？Howtogetthetruesizeofaninclinedplane?换面法Projectiontransformation：物体本身在空间的位置不动，而用某一新投影面（辅助投影面）代替原有投影面，使物体相对新的投影面处于解题所需要的有利位置，然后将物体向新投影面进行投射。Withtheobjectstayingwhereitis,usinganewprojectionplane(auxiliaryprojectionplane)toreplacetheoldprojectionplanesuchthattheproblemcanbesolvedbythenewprojectionoftheobject解决方法：更换投影面。Solutions:AuxiliaryprojectionplaneVHABa

b

ab二、新投影面的选择原则Auxiliaryprojectionplanesetting1.新投影面必须对空间物体处于最有利的解题位置。Thenewprojectionplanemustbeplacedpurposelyinordertosolvetheproblem平行于新的投影面Paralleltothenewprojectionplane垂直于新的投影面Perpendiculartothenewprojectionplane2.新投影面必须垂直于某一保留的原投影面，以构成一个相互垂直的两投影面的新体系。Thenewprojectionplanemustbeperpendiculartooneoldprojectionplanewhichiskeptsothattheycanconstituteanewperpendicularprojectionsystem.Pa1b1VHAa

a

axX

⒈更换一次投影面Firstprojectiontransformation旧投影体系Oldsystem

X—VH新投影体系NewsystemP1HX1—A点的两个投影：a，a

TwoprojectionsofpointA:a，a

A点的两个投影：a，a1TwoprojectionsofpointA:a，a1⑴新投影体系的建立

Theconstructionofnewprojectionsystem三、点的投影变换规律TheruleofpointprojectiontransformationX1P1a1ax1

VHXP1HX1a

aa1

axax1.ax1

VHXP1HX1a

aa1VHA

a

axXX1P1a1ax1

⑵新旧投影之间的关系Therelationshipofoldandnewprojections

aa1

X1

a1ax1=a

ax

点的新投影到新投影轴的距离等于被代替的投影到原投影轴的距离。axa

一般规律：

点的新投影和与它有关的原投影的连线，必垂直于新投影轴。.

XVHaa

ax更换H面replacingH⑶求新投影的作图方法Drawingofnewprojection

VHXP1HX1由点的不变投影向新投影轴作垂线，并在垂线上量取一段距离，使这段距离等于被代替的投影到原投影轴的距离。Drawlineperpendiculartonewaxisfromtheunchangedprojection,extendthelinesamedistanceasfromthereplacedprojectiontotheoldaxisaa

X1P1H

a1axax1ax1更换V面replacingV●a1作图规律：

..VHABa

b

ab四、直线的换面Projectiontransformationofline把一般位置直线变换成投影面平行线Turningobliquelineintolineparalleltoaprojectionplane用P1面代替V面，在P1/H投影体系中，AB//P1。ReplacingVwithP1,AB//P1X1HP1P1a1b1空间分析：

换H面行吗？DoesreplacingHwork?不行！No!作图：例：求直线AB的实长及与H面的夹角。Eg.FindthetruelengthandofABa

b

abXVH新投影轴的位置？Thepositionofnewaxis?a1●b1●与ab平行Paralleltoab

.把面平行线变换成投射线Turningparallellineintoperpendiculartoaprojectionplane可以通过投影面变换换将一般位置直线转化为面平行线，将面平行线转换为投射线.Now,obliquelinecanbeturnedintoparallellineandparallellinecanbeturnedintoperpendiculartoprojectionplane.问：是否可将一般位置直线转换为投射线？Question:Canobliquelinebeturnedintoperpendiculartoprojectionplane?答:用一次换面不可以.(为什么?)Answer:Onlyonetransformationcannotmakeit.(Why?)可用二次换面实现.Withsecondauxiliaryprojectionplane二次换面Secondaryauxiliaryview3.更换两次投影面Secondauxiliaryprojectionplane先把V面换成平面P1，P1H，得到中间新投影体系:FirstlyV->P1andletP1H，sogetthefirstnewprojectionsystem:P1HX1—再把H面换成平面P2，P2

P1，得到新投影体系:ThenH->P2andletP2

P1,getthesecondnewprojectionsystem:X2—P1P2⑴新投影体系的建立Constructionofnewprojectionsystem按次序更换TransforminginsequenceAaVHa

axXX1P1a1ax1

P2X2

ax2a2

ax2

a

aXVH

⑵求新投影的作图方法Stepsofdrawing

a2X1HP1X2P1P2

作图规律

a2a1X2轴a2ax2=aax1a1

axax1

..把一般位置平面变换成投影面垂直面Transformingobliqueplaneintoplaneperpendiculartoprojectionplane如果把平面内的一条直线变换成新投影面的垂直线，那么该平面则变换成新投影面的垂直面。Ifalineinoneplanecanbetransformedintoalineperpendiculartonewprojectionplane,thatplanethenhasbeentransformedintoplaneperpendiculartonewprojectionplane.空间分析Spatialanalysis：两平面垂直需满足什么条件？Whenaretwoplanesperpendiculartoeachother?五、平面的换面Projectiontransformationofplane一般位置直线变换成投影面垂直线，需经几次变换？Howmanytransformationsshouldbeneededtoturnoneobliquelineintooneperpendiculartoprojectionplane?

a

b

c

abcdVHABCDX

d

P1X1c1b1a1

d1在平面内取一条投影面平行线，经一次换面后变换成新投影面的垂直线，则该平面变成新投影面的垂直面。ConstructonehorizontallineintheplaneandturnitintooneperpendiculartothenewVerticalplaneafteronetransformation,andtheplanebeperpendiculartothenewVerticalplane.能否只进行一次变换？Coulditbedonebyonlyonetransformation?思考：若变换H面，需在面内取什么位置直线？WhatkindoflineshouldbedrawnintheplaneifonenewHorizontalplaneneeded?正平线！Frontalline!αa

b

c

acbXVH例：把三角形ABC变换成投影面垂直面。TurnthetriangleABCintoplaneperpendiculartoprojectionplaneHP1X1★在平面内取一条水平线AD。DrawahorizontallineADinABCd

d★将AD变换成新投影面的垂直线。TurnADintolineperpendiculartonewprojectionplaneb1●a1d1●c1●

α反映平面对哪个投影面的夹角？αistheanglebetweenABCandwhichplane?.2、将投射面换为面平行面Turnperpendicularplaneintoparalleltoprojectionplane能否将一般位置平面换为面平行面？Canweturnanobliqueplaneintooneparalleltoaprojectionplane?b

a

c

abcb1

c1

a1

ABC真形例：把三角形ABC变换成投影面平行面。TurningTriangleABCintoplaneparalleltoaprojectionplaneXVHa1

b1●一次换面,把一般位置平面变换成新投影面的垂直面；Firsttransformationofprojectionplaneturnstheobliqueplaneintoperpendicularone.二次换面，再变换成新投影面的平行面。Secondtransformationofprojectionplaneturnstheperpendicularplaneintoparallelone.X2P1P23.把一般位置平面变换成投影面平行面Turnanobliqueplaneintooneparalleltoprojectionplaneaba

c

b

XVHc作图Procedure：AB是水平线AB∥Ha2●c2●b2●c1●X2轴的位置？PositionofaxisX2?

平面的实形Trueshapeoftheplane.X1HP1.与其平行Paralleltothisb1′

距离Distancedd1′X1HV1X2V1H2c2

d

例1：求点C到直线AB的距离，并求垂足D。FindthedistanceoffrompointCtolineABandindicatetheperpendicularfootDcc

b

a

abXVH

如下图：当直线AB垂直于投影面时，CD平行于投影面，其投影反映实长。Asthefigurebelowshows:IfAB⊥P,thenCD∥P,anditsprojectionofPisthetruelength.APBDCca

b

d求C点到直线AB的距离，就是求垂线CD的实长。TherequireddistanceisthetruelengthofperpendicularlineCDc1′

a1′

a2

b2

d2

过作线平行于x2轴。Drawlineparalleltox2throughc1′...如何确定d1′点的位置？Howtodeterminethepositionofpointd1′?六、换面法的应用Applicationoftransformationofprojectionplaneb

a

abcd●c

例2：已知两交叉直线AB和CD的公垂线的长度为MN，且AB为水平线，求CD及MN的投影。Given:ThelengthofcommonperpendicularofskewlinesABandCDisMN,AB∥HReq’d:CompletetheprojectionsofCDandMNMN●m

●d

●a1′b1′m1′●n1′●c1′●d1′●nVHXHV1X1圆半径=MNTheradiusofcircle=MN●n

●m当直线AB垂直于投影面时，MN平行于投影面，这时它的投影m1n1=MN,且m1n1⊥c1d1。P1ACDNMc1d1a1m1b1n1B请注意各点的投影如何返回？求m点是难点。..AB与CD都平行于投影面时，其投影的夹角才反映真形（60°），因此需将AB与C点所确定的平面变换成投影面平行面。Theangleistruesize(60°)whenbothABandCDareparalleltoprojectionplane.SoturntheplanedeterminedbyABandCintooneparalleltoprojectionplane.例3：过C点作直线CD与AB相交成60º角。ConstructlineCD,requiredtheanglebetweenCDandABis60ºd

X1HV1X2V1H2ab

a

c

b

XVHcc2●●●c1′●a1′

b1′●a2●d2●d●b2●

几个解？Howmanysolutions?两个解！Two!60°D点的投影如何返回？HowtogobacktheprojectionsofpointD?..H2V1X2HV1X1c

d

b

a

dacb●d1′●c1′●a1′●d2●b1′c2●●a2≡

b2θVHXθ例4：求平面ABC和平面ABD之间的夹角。FindtheanglebetweenplaneABCandABD在投影图中,两平面的交线垂直于投影面时，则两平面垂直于该投影面，它们的投影积聚成直线，直线间的夹角为所求。Iftheintersectinglineoftwoplanesisperpendiculartoaprojectionplane,bothofplanesareaccumulatedtotwolineswhichtheanglebetweenthemistheangl