XXXX秋季第講換面機械制圖北京航空航天大學出社

例1：求已知直線與平面的夾角的實際大小。Findthetrueanglebetweenthegivenlineandplanem'

n'

nm步驟：求MN與△ABC的交點K(求交)PiercingpointofMNand△ABCa'c'b'abc1'122'

d'

dPvk'

k??過M作MD△ABCDrawMD△ABC求MD與△ABC的交點S(求交)PiercingpointofMDand△ABC求MKL(△MKL)的實形即(分別求出MS、KS的實長)ThetruesizeofMKL(△MKL)is

MKSs'

s先求得KS、MS

的實長FindthetruelengthofKSandMSMPNSKθ例1：求已知直線與平面的夾角的實際大小。

Findthetrueanglebetweenthegivenlineandplanem'

n'

nm步驟：a'c'b'abc1'122'

d'

de'

e??過M作MD△ABCDrawMD△ABCNDMEMN求NMD(△MND)的實形TruesizeofNMD(△MND)NMD的余角即為ThecomplementaryangelofNMDis需求得MN、ND、MD的實長如畫水平線NE，只需求MN、ME的實長MPNSKθDE不求交？Withoutfindingtheintersection?問題：如何求異面直線的公垂線？Problem：Howtoconstructcommonperpendiculartoskewlines?如果其中一條直線是投射線？Whatifoneoftheskewlinesperpendiculartoprojectionplane?第四章投影變換Projectiontransformation一、問題的提出Problems如何求一般位置平面圖形的真實大?。縃owtogetthetruesizeofaninclinedplane?換面法Projectiontransformation：物體本身在空間的位置不動，而用某一新投影面（輔助投影面）代替原有投影面，使物體相對新的投影面處于解題所需要的有利位置，然后將物體向新投影面進行投射。Withtheobjectstayingwhereitis,usinganewprojectionplane(auxiliaryprojectionplane)toreplacetheoldprojectionplanesuchthattheproblemcanbesolvedbythenewprojectionoftheobject解決方法：更換投影面。Solutions:AuxiliaryprojectionplaneVHABabab二、新投影面的選擇原則Auxiliaryprojectionplanesetting1.新投影面必須對空間物體處于最有利的解題位置。Thenewprojectionplanemustbeplacedpurposelyinordertosolvetheproblem平行于新的投影面Paralleltothenewprojectionplane垂直于新的投影面Perpendiculartothenewprojectionplane2.新投影面必須垂直于某一保留的原投影面，以構成一個相互垂直的兩投影面的新體系。Thenewprojectionplanemustbeperpendiculartooneoldprojectionplanewhichiskeptsothattheycanconstituteanewperpendicularprojectionsystem.Pa1b1VHAaaaxX⒈更換一次投影面Firstprojectiontransformation舊投影體系Oldsystem

X—VH新投影體系NewsystemP1HX1—A點的兩個投影：a，aTwoprojectionsofpointA:a，aA點的兩個投影：a，a1TwoprojectionsofpointA:a，a1⑴新投影體系的建立

Theconstructionofnewprojectionsystem三、點的投影變換規(guī)律TheruleofpointprojectiontransformationX1P1a1ax1VHXP1HX1aaa1axax1.ax1VHXP1HX1aaa1VHA

aaxXX1P1a1ax1⑵新舊投影之間的關系Therelationshipofoldandnewprojections

aa1

X1a1ax1=aax點的新投影到新投影軸的距離等于被代替的投影到原投影軸的距離。axa一般規(guī)律：點的新投影和與它有關的原投影的連線，必垂直于新投影軸。.XVHaaax更換H面replacingH⑶求新投影的作圖方法DrawingofnewprojectionVHXP1HX1由點的不變投影向新投影軸作垂線，并在垂線上量取一段距離，使這段距離等于被代替的投影到原投影軸的距離。Drawlineperpendiculartonewaxisfromtheunchangedprojection,extendthelinesamedistanceasfromthereplacedprojectiontotheoldaxisaaX1P1Ha1axax1ax1更換V面replacingV●a1作圖規(guī)律：..VHABabab四、直線的換面Projectiontransformationofline把一般位置直線變換成投影面平行線Turningobliquelineintolineparalleltoaprojectionplane用P1面代替V面，在P1/H投影體系中，AB//P1。ReplacingVwithP1,AB//P1X1HP1P1a1b1空間分析：

換H面行嗎？DoesreplacingHwork?不行！No!作圖：例：求直線AB的實長及與H面的夾角。Eg.FindthetruelengthandofABababXVH新投影軸的位置？Thepositionofnewaxis?a1●b1●與ab平行Paralleltoab.把面平行線變換成投射線Turningparallellineintoperpendiculartoaprojectionplane可以通過投影面變換換將一般位置直線轉化為面平行線，將面平行線轉換為投射線.Now,obliquelinecanbeturnedintoparallellineandparallellinecanbeturnedintoperpendiculartoprojectionplane.問：是否可將一般位置直線轉換為投射線？Question:Canobliquelinebeturnedintoperpendiculartoprojectionplane?答:用一次換面不可以.(為什么?)Answer:Onlyonetransformationcannotmakeit.(Why?)可用二次換面實現(xiàn).Withsecondauxiliaryprojectionplane二次換面Secondaryauxiliaryview3.更換兩次投影面Secondauxiliaryprojectionplane先把V面換成平面P1，P1H，得到中間新投影體系:FirstlyV->P1andletP1H，sogetthefirstnewprojectionsystem:P1HX1—再把H面換成平面P2，P2P1，得到新投影體系:ThenH->P2andletP2P1,getthesecondnewprojectionsystem:X2—P1P2⑴新投影體系的建立Constructionofnewprojectionsystem按次序更換TransforminginsequenceAaVHaaxXX1P1a1ax1P2X2ax2a2ax2aaXVH⑵求新投影的作圖方法Stepsofdrawinga2X1HP1X2P1P2

作圖規(guī)律

a2a1X2軸a2ax2=aax1a1axax1..把一般位置平面變換成投影面垂直面Transformingobliqueplaneintoplaneperpendiculartoprojectionplane如果把平面內的一條直線變換成新投影面的垂直線，那么該平面則變換成新投影面的垂直面。Ifalineinoneplanecanbetransformedintoalineperpendiculartonewprojectionplane,thatplanethenhasbeentransformedintoplaneperpendiculartonewprojectionplane.空間分析Spatialanalysis：兩平面垂直需滿足什么條件？Whenaretwoplanesperpendiculartoeachother?五、平面的換面Projectiontransformationofplane一般位置直線變換成投影面垂直線，需經幾次變換？Howmanytransformationsshouldbeneededtoturnoneobliquelineintooneperpendiculartoprojectionplane?

a

b

cabcdVHABCDX

dP1X1c1b1a1d1在平面內取一條投影面平行線，經一次換面后變換成新投影面的垂直線，則該平面變成新投影面的垂直面。ConstructonehorizontallineintheplaneandturnitintooneperpendiculartothenewVerticalplaneafteronetransformation,andtheplanebeperpendiculartothenewVerticalplane.能否只進行一次變換？Coulditbedonebyonlyonetransformation?思考：若變換H面，需在面內取什么位置直線？WhatkindoflineshouldbedrawnintheplaneifonenewHorizontalplaneneeded?正平線！Frontalline!αab

cacbXVH例：把三角形ABC變換成投影面垂直面。TurnthetriangleABCintoplaneperpendiculartoprojectionplaneHP1X1★在平面內取一條水平線AD。DrawahorizontallineADinABCdd★將AD變換成新投影面的垂直線。TurnADintolineperpendiculartonewprojectionplaneb1●a1d1●c1●

α反映平面對哪個投影面的夾角？αistheanglebetweenABCandwhichplane?.2、將投射面換為面平行面Turnperpendicularplaneintoparalleltoprojectionplane能否將一般位置平面換為面平行面？Canweturnanobliqueplaneintooneparalleltoaprojectionplane?bacabcb1c1a1ABC真形例：把三角形ABC變換成投影面平行面。TurningTriangleABCintoplaneparalleltoaprojectionplaneXVHa1b1●一次換烈面,母把紅一般位案置平面西變換成建新投影囑面的垂背直面；Firs提t(yī)tr洲ansf暢orma撲tion絡of灰proj晝ecti誼onp憶lane畝tur殊nst拼heo咸bliq閱uep興lane兆int蒼ope振rpen樓dicu筒lar紀one.二次換修面，再個變換成扶新投影選面的平鮮行面。Seco泊ndt令rans斧form盟atio芳nof臭pro吃ject齊ion蝴plan鹿etu癥rns琴the跟perp崇endi適cula斧rpl異ane字into硬par乖alle勢lon算e.X2P1P23.把搖一般位置圍平面變換膀成投影面占平行面Turn秒an能obli領que閣plan牲ein疤too碰nep顯aral叛lel蕉top何roje圈ctio蓮npl問aneabacbXVHc作圖Proc障edur結e：AB是水平淺線AB∥鐮Ha2●c2●b2●c1●X2軸的位置？Pos遍iti墓on俘of冰axi幫sX2?平面的蓄實形Tru批es角hap鴿eo字ft湯he遇pla莖ne.X1HP1.與其平要行Par組all距el墓to鞏thi聽sb1′距離Dis哈tan茅cedd1′X1HV1X2V1H2c2d例1：求點C到直線AB的距離，并求垂足D。FindthedistanceoffrompointCtolineABandindicatetheperpendicularfootDccbaabXVH如下圖瘋：當直毛線AB忘垂直于錢投影面座時，C煮D平行宋于投影織面，其貴投影反唉映實長譽。Ast農hef溪igur能ebe另low番show儲s:冷IfA率B⊥P,啄then內CD∥槍P,判and算its村proj監(jiān)ecti葡ono瞞fP誤ist狂het粥rue詳leng證th.APBDCcabd求C點逆到直線煤AB的祝距離，隔就是求西垂線C氧D的實系長。The副requ肉ired款dis銳tanc缸eis缸the朱tru拔ele瞎ngth誕of趟perp情endi兇cula繪rli心neC關Dc1′a1′a2b2d2過作線平徹行于x2軸。Dra冠wl徐ine邀pa運ral類lel宇tox2thro鴨ughc1′...如何確鋒定d1′點的灶位置？How射tod鉗eter勿mine核the筋pos測itio醬nof遙poi隔ntd1′?六、換面緊法的應用App朽l(xiāng)ic濤ati棉on香of忽tra專nsf妻orm字ati麥on言of切pro筍jec各tio傍np蘋lan棚ebaabcd●c例2：后已知兩旱交叉直屯線AB走和CD抖的公垂哈線的長教度為M抄N，且夜AB為垮水平線腳，求C丟D及M唱N的投荒影。Giv嫩en:共Th脖el士eng立th眉of鋪com歷mon呀pe紙rpe柜ndi具cul腔ar混of矛ske譯wl并ine窄sA堵Ba棒nd愁CD坊is長MN急,A友B∥HReq’凝d:C雄ompl急ete加the懂proj糠ecti丈ons刮ofC桃Dan交dMNMN●m●d●a1′b1′m1′●n1′●c1′●d1′●nVHXHV1X1圓半徑弊=MNThe敲radi煉uso齒fci食rcle概=MN●n●m當直線AB垂直于投影面時，MN平行于投影面，這時它的投影m1n1=MN,且m1n1⊥c1d1。P1ACDNMc1d1a1m1b1n1B請注意各紙點的投影行如何返回社？求m點是滔難點。..AB與怕CD都廳平行于配投影面邀時，其均投影的氧夾角才送反映真續(xù)形（6挺0°）腦，因此比需將A困B與C剩點所確批定的平幫面變換禿成投影眠面平行拉面。The損an值gle啟is某tr瘦ue順siz揪e(頭60°貢)w謎hen挨bo洋th莖AB梢and意CD鋒ar騎ep返ara質lle頂lt步op宇roj腔ect杰ion桐pl驢ane武.S政ot范urn片th裕ep液lan嘗ed穗ete悄rmi羊ned犯by莊AB禮an璃dC遮in柔t(yī)o伴one跨pa交ral敢lel廢to忽pr占oje籌cti棕on碰pla令ne.例3：可過C點夫作直線可CD與刑AB相蹤蝶交成6淚0o角貌。Cons梢truc崇tli拖neC動D,r呈equi制red堤the孫angl搬ebe場twee趕nCD蹈and昂AB匹is6鏟0odX1HV1X2V1H2abacbXVHcc2●●●c1′●a1′

b1′●a2●d2●d●b2●幾個解武？How線ma望ny磁sol熊uti點ons競?兩個解箱！Two!60°D點的賄投影如許何返回顆？How變tog雨o