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)ThetruesizeofMKL(△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TruesizeofNMD(△MND)•NMD的余角即为ThecomplementaryangelofNMDis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OldsystemX—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aa1VHAa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Projectiontransformationofline1.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.FindthetruelengthandofABababXVH新投影轴的位置？Thepositionofnewaxis?a1●b1●与ab平行Paralleltoab.2.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，P1H，得到中间新投影体系:FirstlyV-P1andletP1H，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a2a1X2轴a2ax2=aax1a1axax1..1.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abcdVHABCDXd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●a1d1●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。Secondtransformationofprojectionplaneturns