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1、同学们好！大家好!Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.第三章 直线第二节 各种位置直线第一节 直线的投影第三节 一般位置线段的实长及其 对投影的倾角第四节 两直线的相对位置第五节
2、直角投影定理Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.H 直线的投影仍为直线，两点确定一条直线，直线上两点的投影用直线连接，就得到直线的投影。b第一节 直线的投影直线垂直于投影面其投影积
3、聚为一点(积聚性)作CB/ab,CB/ab,则ABCABC为直线ABAB对投影面H H的倾角.直线倾斜于投影面其投影比实长短: ab=ABab=ABcos(类类似性）似性）直线平行于投影面其投影反映线段实长: fg=FGfg=FG（真实性）（真实性）投影特性FABae(d)DEC( (c) )gfGCEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose
4、.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.投影面平行线平行于某一投影面而与其余两投影面倾斜投影面垂直线正平线（只平行于面）侧平线（只平行于面）水平线（只平行于面）正垂线（垂直于面）侧垂线（垂直于面）铅垂线（垂直于面）一般位置直线与三个投影面都倾斜的直线垂直于某一投影面(平行于另两投影面)返回第二节 各种位置直线特殊位置直线一、直线的分类：Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5
5、.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.VHX0ZYW二、相对投影面各种位置直线的投影 一般位置直线的三个投影仍为直线；三个投影都倾斜于投影轴；投影长度小于线段的实长；投影与投影轴的夹角，不反映直线对投影面的倾角。babababababaAB一般位置直线ABXZYHYWObababaXZYWOEvaluation only.C
6、reated with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.直线上点的投影，必在直线的同面投影上；（cab, ca b ,ca b)线段上的点分割线段之比，投影后保持不变。AC/CB=ac/cb= a c / c b = a b/ c b
7、)1.一般位置直线上点的投影VHX0ZYWXYWYHbababaABC是直线AB上的点ABccc0ZbababacccCEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例1 如图中黑色的图形所
8、示，作出分 线段AB为3:2的点c的两面投影。0作法： 1.过点a任作一条直线ae； 2.在ae上截取五等分； 3.连接be； 4.在离点a三等分点处作直线平行于be，交ab上一点c； 5.过点c作直线垂直于ox,交 ab于点c。 点C(c, c)即为所求。ababccexEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .N
9、ET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.VHX0ZYW2.投影面平行线X0ZYWYHbababa1) 正平线（投影面V的平行线AB）AB投影特性：1. ab1. ab / OZ / OZ ， a b/ OX a b/ OX；2. a2. a b b = A B = A B；3 3. .正面投影正面投影反映、角的真实大小。bababaABABABEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Cop
10、yright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.VHX0ZYW投影特性：1.a1.a b b / OX / OX，ab/ OYwab/ OYw；2.a b = A B2.a b = A B；即：水平投影即：水平投影反映线段实长及、角的真实大小。aababbXababba0zYHYW2) 水平线（投影面H的平行线AB）ABEvaluation only.C
11、reated with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.3)侧平线（投影面W的平行线AB）XZabbbaOYHYWaVHX0ZYWaababbAB（小结）投影面平行线的投影特性：1.在直线所平行的投影面上的投影，反映线段实长；它与投影
12、轴的夹角，分别反映直线与另两投影面夹角的真实大小。2.在另外两个投影面上的投影，平行于相应的投影轴，长度缩短。Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.3.投影面垂直线VHX0ZYW1)
13、正垂线(投影面V的垂直线AB)(b)aABbabababa(b)a投影特性：1. a ( b )积聚成一点 2. a b Ox ； a b Oz 3. a b = a b = ABzxyHyHoEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright
14、2004-2011 Aspose Pty Ltd.(投影面H的垂直线AB)投影特性：1、 a b 积聚成一点 2、 a b ox,a b oyw 3、 a b = a b = ABzbxaba(b)oYHYWaVHX0ZYWabbaa(b)AB2) 铅垂线Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Cl
15、ient Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.(投影面 W 的垂直线AB)投影面垂直线的投影特性：1.在与直线垂直的投影面上的投影积聚成一点一点。2.在另外两个投影面上的投影垂直于相应的投影轴且反映线段实长。VHX0ZYWbaaba(b)ABZxa(b)baoYHYWab3) 侧垂线Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation
16、only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.zz1z1zoxababkk12例2 侧平线上的点已知AB的两面投影及其上面的点K的正面投影，求点K的水平投影。利用点在线上分割线段成定比，投影后不变的性质作图。Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Ev
17、aluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.第三节 一般位置线段的实长及其对投影面的倾角 方法是：以线段在某一投影面上的投影长为一直角边，两端点与这个投影面的距离差为另一直角边形成的直角三角形。其斜边是线段的实长，斜边与投影长的夹角就是该直线与这个投影面的倾角。VHX0ZYWbababaAB X0ZYWYHbababa在正面投影上求线段实长与倾角在水平投影上求线段实长与倾角在侧面投影上求线段实长与倾角实长实长实长实长
18、实实长长 直角三角形法BoEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.直角三角形求线段实长及其与投影面的倾角中的三个三角形 设所求线段为AB 在三个直角三角形中，斜边为线段实长；一个直角边
19、为某投影长，该投影与斜边的夹角为该直线与投影面的夹角；夹角所对的边为线段两端点相应的坐标差。AB线段实长ZABab的长AB线段实长XABa b 的长 AB线段实长YABa b 的长Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-20
20、11 Aspose Pty Ltd.例1 已知线段的实长AB，求它的水平投影。aAB 根据已知条件，要求得ab，其方法一是求得A、B两点的Y坐标差(YAB ) ；方法二是求得ab的长。此题有两解(多解时一般只画一解）bxaobABbxaobABbxaobEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Cli
21、ent Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.a bA B方法二 已知线段的实长AB，求它的水平投影。aAB此题有两解bxaob解法二Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyrigh
22、t 2004-2011 Aspose Pty Ltd.方法三：求出ab的长ABbxaobABxabABBaZAB12以AB两点的Z坐标差为一直角边的直角三角形中，另一直角边为AB水平投影ab的长。即：12=abEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Co
23、pyright 2004-2011 Aspose Pty Ltd.ZAB例例2 2 已知直线已知直线ABAB的的=30=30 求作求作ABAB的正面投影。的正面投影。oxababZAB301.分析要求得ab，其实就是求b；要求得b也就是想办法找到A、B两点的Z坐标差或者求出ab的长。B02.作图1）作abB0,使= 30A的对边为ZAB2）过a作直线平行于Ox轴，与过b而垂直于ox轴的直线交于一点b0 3）以b0为圆心以ZAB为半径画圆弧，交bb0的延长线于点b4）用直线连接ab即为所求。b0Evaluation only.Created with Aspose.Slides for .NET
24、 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.oxaabkbcd6820dc6820k例3 已知CDAB=K，CDH， 求CD的正面投影。求k同前。由于CDH所以cdox轴。Evaluation only.Created with Aspose.Slides for .NET 3.5 Cli
25、ent Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd. 同学们好！Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation
26、only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.第四节第四节 两直线的相对位置两直线的相对位置 分析： 1.已知ABCD,根据正投影图的作图法可知： (AaP BbP CcP DcP) P面，则平面AB bPaP CDdPcP； 2.两个平行平面（AB bPaP与CDdPcP）与第三平面（P可被看作是H、V、W面）相交，交线平行； 3.综上所述，我们可以得出：若空间两直线平行那么他们的同面投影也对应平行（如：ab cd、ab cd、ab
27、cd） 一.两平行直线XZYWOYHXZYWOBDabababcddcdcPAaPbPCcPdPAXYHXZYWOabababcddcdcEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例1
28、判断两直线是否平行aabdcbcdox方法一： 看直线的方位：由投影图可以看出ab、cd同向（均由后向前）；a b 与c d 反向（a b 由上向下，c d 由下向上）由此看出AB与CD不平行。方法二： 看比例：a b /c d 1,ab/cd1,由此可知，AB不平行于CD。Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .
29、NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例2 如图所示，判断两侧平线的相对位置。bX0acddcba作辅助直线AD与BC的两面投影,判断：AD与BC是两相交直线,则AB与CD共面. 只要证明AB与CD共面则有ABCD。X0abccbabcdc c cabcbcaabcbcdYWX0abccbabcdc c cabcbcaabcbcdYWX0abccabcdc c cabccaabccdYWX0abccacdcabcaabdcdYWYHaZ方法三：求第三投影(三个投影都互相平行，则ABCD)方法四：看已知
30、的二 直线是否共面Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.abcdc a b d 例3 判断图中两条直线是否平行。 由此可知：对于一般位置直线，只要有两个同名投影互相平行，空间两直线就
31、平行。即即AB/CD返回1 1 a b : c d = a b: c d a b c d , a b c d a b 1 1 c d ; a b 1 1 c d; a 1 : 1 d = a 1: 1 da a 1 1 d d AB与CD共面a b c d , a b c doxEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for
32、 .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.二、两相交直线因此，三对同面投影都相交，且交点符合点的三面投影特规律ZYHabX0YWbbaaeeedcdcdcPABCDEapbpcpdpe两直线相交有且仅有一个交点（E）交点是相交两直线的公有点（P面可以看作H、V、W面）Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evalu
33、ation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例1 已知AB与CD相交，又知AB的两面投影及CD的正面投影c d ，且CD平行于V面。求CD的水平投影a b c d aboxk 分析：ABCD=KKAB，KCDk ab，k cda k /k b =ak/kb又因为AB平行于W面，所以cd/ox轴。用定比分点法求k。解：1.过a任作一条直线ae；2.在ae上截取a1=a b ； a2= a k 的长度3.连接1b4.作2k1b
34、,与ab交于点k5.过k作直线平行于ox轴6.过c 、 d 作直线垂直于ox,交上述直线于点c、de12a k 的的长长cdEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.三、交错直线交错两直
35、线的投影既不符合平行两直线的投影特性，也不符合相交两直线的投影特性。X X X XYHX X XXbaaeeccfbaaeeccfbaaeeccfbaaeccfbaaeccfbaaeccfbaaeccfbaaeccfbaaeccfbaacc AB与CD交错，为什么他们在P面上的投影相交呢？因为直线AB上的点F和直线CD上的点E位于指向P面的同一条投射线上，所以点F、E在P 面上的投影重合，点F的投影可见而点E的投影不可见。ZYH0YWbabeddcdf0YWabeddcdfbabeddcdf0YWabeddcdf0YWabeddcdf0YWbabeddcdf0YWabeddcdfbabeddc
36、df0YWabeddcdf0YWeddcdfd ddpbp(f )ABDapFEfp（ep）PScpfaZYH0YWbbabeddcdf0YWbabeddcdfbbabeddcdf0YWbabeddcdf0YWbabeddcdf0YWbbabeddcdf0YWbabeddcdfbbabeddcdf0YWbabeddcdf0YWbeddcdfd daaeefaaeefaaeefaaefaaefaaefaaefaaefaaefaac AB与CD交错，为什么他们在P面上的投影相交呢？因为直线AB上的点F和直线CD上的点E位于指向P面的同一条投射线上，所以点F、E在P 面上的投影重合，点F的投影可见，
37、本题中点E、F是对V面的一对重影点；点E在前，所以点F的正面投影f 不可见。(f )CEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例1 判断两直线的相对位置xaacddcbbodacbyz
38、y 解法1:作出侧面投影,由三面投影可知,两直线交错Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.xaacddcbbo例2 判断两直线的相对位置(解法2)11 用点分割直线段之比，投影后保持
39、不变的性质判断。 由水平投影可判断，点不属于直线AB，故两直线交错。Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.第五节 直角投影定理 直角投影定理： 互相垂直的两直线，当一边为某投影面的平
40、行线时，他们在该投影面上的投影是直角。 逆定理也是成立的。 AB BC，直角边 AB平行于投影面P(P面 可以被看作是H、V、W面） AB P，Bbp P ABBbp又 ABBC， ABQ，即AB bpcp ab AB, ab Q, 即a pb pb pcpabcX0acbPABCQapbpcpQapbpcpEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with As
41、pose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.直角投影定理：交错二直线互相垂直的两直线，当一边为某投影面的平行线时，他们在该投影面上的投影是直角。Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3
42、.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.bb例1 作直线AB垂直于直线EF。11Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose
43、 Pty Ltd. f 例2 作直线EF，使其垂直于直线AB和直线CD。eefEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.boxabk1akYK1YK例3 求点K到直线AB的距离（投影和实
44、长）。分析1.abox,ABV则有k1ab2.用直角三角形法求出K1的实长便得解。Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例4 作三角形ABC，ABC为直角，使BC在MN上，且BC:A
45、B=2:3。mxmnaonabbccMN平行于投影面V，BC在MN上，直角边BC平行于投影面V，又因为ABC=90，所以有：b c =BC,a b b c 用直角三角形法求线段AB实长| YA-B |ABa bbc的长2份3份Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.
46、2.0.0.Copyright 2004-2011 Aspose Pty Ltd.例5 过点A作等边三角形ABC，使B、C两点属于直线KM。dAD实长oxd边 实 长22.122.130kbabacmmkcYADYAD分析：1.KMV面 cb=BC adkm2.只要求得过顶 点A的高，则ABC可作出；也就是说ABC的边长就求出来了，即求得了cbEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation on
47、ly.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.obab75ee75ab的长cca例6 作等腰直角ABC,使AC=AB，C点在AE上。xEvaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.Evaluation only.Created with Aspose.Slides for .NET 3.5 Client Profile 5.2.0.0.Copyright 2004-2011 Aspose Pty Ltd.
