画法几何直线直线的相对位置直角投影定理直角三角性法.ppt

同学们好！ 大家好! Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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 直角投影定理 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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聚为一点(积聚性) 作CB/ab,则ABC为直线AB对投影面H的倾角. 直线倾斜于投影面其投影比实长短: ab=ABcos(类类似性） 直线平行于投影面其投影反映线段实长: fg=FG（真实性） 投影特性 FA B a e(d) D E C (c) g f G C Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. V H X0 Z Y W 二、相对投影面各种位置直线的投影二、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。 b a b a b a b a b a b a A B 一般位置直线AB X Z YH YWO b a b a b a X Z YWO Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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) 1.一般位置直线上点的投影 V H X0 Z Y W XYW YH b a b a b a A B C是直线AB上的点 A B c c c 0 Z b a b a b a c c c C Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 例1 如图中黑色的图形所示，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。 a b a b c c e x Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. V H X0 Z Y W 2.投影面平行线 X 0 Z YW YH b a b a b a 1) 正平线（投影面V的平行线AB） A B 投影特性： 1. a“b“ / OZ ， a b/ OX； 2. a b = A B； 3.正面投影反映、角的真实大小。 b a b a b aA B A B A B Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. V H X0 Z Y W 投影特性： 1.a b / OX，a“b“/ OYw； 2.a b = A B； 即：水平投影反映线段实长及、角的真实大小。 a a b a b b X abab b a 0 z YH YW 2) 水平线（投影面H的平行线AB） A B Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 3)侧平线（投影面W的平行线AB） X Z a b b b a O YH YW a V H X0 Z Y W a a b a b b A B （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.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 3.投影面垂直线 V H X0 Z Y W 1)正垂线(投影面V的垂直线AB) (b) a A B b a b a b a ba (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 = AB z x yH yH o Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 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 = AB z b x a b a(b) o YH YW a V H X0 Z Y W a b b a a(b) A B 2) 铅垂线 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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。 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。 V H X0 Z Y W b a ab a(b ) A B Z x a(b) b a o YH YW a b 3) 侧垂线 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. z z1 z1 z o x a b a b k k 1 2 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.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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。 V H X0 Z Y W b a b a b a A B X 0 Z YW YH b a b a b a 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 直角三角形法 Bo Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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 ZAB ab的长 AB线段实长 XAB a b 的长 AB线段实长 YAB a b 的长 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 例1 已知线段的实长AB，求它的水平投影。 a AB 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） b x a o b AB b x a o b AB b x a o b Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. a b A B 方法二 已知线段的实长AB，求它的水平投影。 a AB 此题有两解 b x a o b 解法二 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 方法三：求出ab的长 AB b x a o b AB x a b AB B a ZAB 1 2 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=ab Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. ZAB 例2 已知直线AB的=30 求作AB的正面投影。 o x a b a b ZAB 30 1.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。 B0 2.作图 1）作abB0,使= 30 A的对边为ZAB 2）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b 4）用直线连接ab 即为所求。 b0Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. o x a a b k b c d 68 20 d c 68 20 k 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.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 同学们好！ Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 第四节 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 cd） 一.两平行直线 X Z YW O YH X Z YW O BD a b a b a b c d d c d c P A aP bP C cP dP A X YH X Z YW O a b a b a b c d d c d c Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 例1 判断两直线是否平行 a a b d c b c d o x 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.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. 例2 如图所示，判断两侧平线的相对位置 。 b X 0 a c d d c b a 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。 X 0 a b c c b a b c d c c c a b c b c a a b c b c d YW X 0 a b c c b a b c d c c c a b c b c a a b c b c d YW X 0 a b c c a b c d c c c a b c c a a b c c d YW X 0 a b c c a c d c a b c a a b d c d YW YH a Z 方法三： 求第三投影(三个投影都 互相平行，则ABCD) 方法四： 看已知的二 直线是否共面 Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. a b c d c 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 直线就平行。 即AB/CD 返回 1 1 ab: cd = a b: c d ab cd, a b c d ab1 1cd; a b 1 1 c d; a1: 1d = a 1: 1 d aa 11 d d AB与CD共面 ab cd, a b c d ox Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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 Z YH a b X 0 YW b b a a e e e d c d c d c P A B C D E ap bp cp dp e 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.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.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 a b ox k 分析： ABCD=K KAB，KCD k ab，k cd a 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1b 4.作2k1b,与ab交于点k 5.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、d e 1 2 a k 的长 c d Evaluation only.Evaluation only. Created with Aspose.Slides for .NET 3.5 Client Profile .Created with Aspose.Slides for .NET 3.5 Client Profile . Copyright 2004-2011 Aspose Pty Ltd.Copyright 2004-2011 Aspose Pty Ltd. Evaluation only.Evaluation only. Created with Aspose.Slides for .NET