畫法幾何：第三講 直線的投影

第三講 直線的投影Projection of Line直線的投影直線的投影 Projections of Lines特殊位置的直線和一般位置的直線特殊位置的直線和一般位置的直線Special position lines&General Position lines直線上的點(diǎn)直線上的點(diǎn)Points on Lines直線的實(shí)長(zhǎng)直線的實(shí)長(zhǎng) True length of a line直線平行性垂直性和直線平行性垂直性和 Parallelism&Perpendicularity of linesaa a b b b 兩點(diǎn)確定一條直線，將兩點(diǎn)兩點(diǎn)確定一條直線，將兩點(diǎn)的同名投影用直線連接，就得到直的同名投影用直線連接，就得到直線的同名投影。線的同名投影。(Projections of 2 points determine the projection of line.)直線對(duì)一個(gè)投影面的投影特性直線對(duì)一個(gè)投影面的投影特性(To single projection plane)一、直線的投影特性一、直線的投影特性CharacteristicsABab直線垂直于投影面直線垂直于投影面投影重合為一點(diǎn)投影重合為一點(diǎn) 積聚性積聚性Point-wise projection直線平行于投影面直線平行于投影面投影反映線段實(shí)長(zhǎng)投影反映線段實(shí)長(zhǎng)存真性存真性 ab=ABTrue length projection直線傾斜于投影面直線傾斜于投影面投影比空間線段短投影比空間線段短 ab=ABcosForeshorten projectionABabAMBabm2 2、直線在三個(gè)投影面中的投影特性直線在三個(gè)投影面中的投影特性Lines projected to the 3-projection-planeLines projected to the 3-projection-plane投影面平行線投影面平行線Lines parallel to a Projection plane平行于某一投影面而平行于某一投影面而與其余兩投影面傾斜與其余兩投影面傾斜投影面垂直線投影面垂直線Lines perpendicular to a projection plane正平線正平線Frontal lines （面）面）側(cè)平線側(cè)平線Profile lines （面）面）水平線水平線Horizontal lines（面）面）正垂線正垂線（W-perpendicular lines）（面）面）側(cè)垂線側(cè)垂線（W-perpendicular lines）（面）面）鉛垂線鉛垂線（H-perpendicular lines）（面）面）一般位置直線一般位置直線General Position lines與三個(gè)投影面都傾斜的直線與三個(gè)投影面都傾斜的直線Inclined to all three projection planes統(tǒng)稱特殊位置直線統(tǒng)稱特殊位置直線(Special position lines)垂直于某一垂直于某一投影面投影面b a aba b b aa b ba 投影面平行線投影面平行線Lines parallel to a Projection plane 在其平行的那個(gè)投影面上的投影反映實(shí)長(zhǎng)，并反映直線與在其平行的那個(gè)投影面上的投影反映實(shí)長(zhǎng)，并反映直線與另兩投影面傾角的實(shí)大。另兩投影面傾角的實(shí)大。When a line is parallel to a projection plane,its projection upon that plane will be the true length of the line.另兩個(gè)投影面上的投影平行于相應(yīng)的投影軸。另兩個(gè)投影面上的投影平行于相應(yīng)的投影軸。Its projection on the adjacent planes will be parallel to the projection axis.水平線水平線Horizontal line側(cè)平線側(cè)平線Profile line正平線正平線Frontal line投投 影影 特特 性：性：與與H面的夾角面的夾角（Angle with H）:與與V面的角面的角:與與W面的夾角面的夾角:實(shí)長(zhǎng)實(shí)長(zhǎng)（True length）實(shí)長(zhǎng)實(shí)長(zhǎng)實(shí)長(zhǎng)實(shí)長(zhǎng)ba aa b b 投影面垂直線投影面垂直線Lines perpendicular to a projection planeLines perpendicular to a projection plane鉛垂線鉛垂線H-perpendicular line正垂線正垂線V-perpendicular line側(cè)垂線側(cè)垂線W-perpendicular line 另外兩個(gè)投反映線段實(shí)長(zhǎng)。且垂直于相應(yīng)的投另外兩個(gè)投反映線段實(shí)長(zhǎng)。且垂直于相應(yīng)的投影軸。影軸。Its projection on the adjacent planes will be perpendicular to the corresponding projection axis and will represent the true length.在其垂直的投影面上，在其垂直的投影面上，投影有積聚性投影有積聚性。投影特性投影特性:c(d)cdd c a b a(b)a b e f efe(f)When a line is perpendicular to a projection plane,it appears as a point upon that plane.一般位置直線一般位置直線General position linesGeneral position lines投影特性：投影特性：三個(gè)投影都縮短。即三個(gè)投影都縮短。即:都不反映空間線段的實(shí)長(zhǎng)及都不反映空間線段的實(shí)長(zhǎng)及與三個(gè)投影面夾角的實(shí)大，與三個(gè)投影面夾角的實(shí)大，且與三根投影軸都傾斜。且與三根投影軸都傾斜。All three projections are foreshortened.That is:Its true length and angles will not be represented in any of these projections.abb a b a 二、直線上的點(diǎn)二、直線上的點(diǎn) Points on lines 若點(diǎn)在直線上若點(diǎn)在直線上,則則點(diǎn)的投影必在直線的同點(diǎn)的投影必在直線的同名投影上。名投影上。If a point in space lies on a line,the projections of that point also lie on the corresponding projections of the line.判別方法判別方法：ABCVHbcc b a a點(diǎn)點(diǎn)C不不在在直線直線AB上上(No)例例1：判斷點(diǎn)：判斷點(diǎn)C是否在線段是否在線段AB上。上。Determine if the point C on line ABabca b c c abca b 點(diǎn)點(diǎn)C在直在直線線AB上上(Yes)例例2：判斷點(diǎn)：判斷點(diǎn)K是否在線段是否在線段AB上。上。Determine if the point K on line ABa b k 因因k 不在不在a b 上，上，故點(diǎn)故點(diǎn)K不在不在AB上。上。k is not on a b,then K lies on line AB.abka b k 當(dāng)直線的投影垂直于投影軸時(shí)有例外出當(dāng)直線的投影垂直于投影軸時(shí)有例外出現(xiàn)，這種情況可增加第三投影。現(xiàn)，這種情況可增加第三投影。Exception may occur to special position line.A profile projection is added in this case.c cc c定比性判斷：平行投影中直定比性判斷：平行投影中直線上的點(diǎn)把各個(gè)投影分割成相同的上的點(diǎn)把各個(gè)投影分割成相同的比例比例 By proportionality:in parallel projection,point on a line divides all projections of the line by same proportion.點(diǎn)C屬于 直線AB，已知C點(diǎn)的V投影，求H投影三、一般位置直線的實(shí)長(zhǎng)與傾角True length and angle of a general lineOXHVa aaxbbbx傾角傾角:直線與在某一投影面上的投影間的夾角為直線與該面的傾角直線與在某一投影面上的投影間的夾角為直線與該面的傾角 Angle between line and its projection will be its angle with that projection plane.Angle between line and its projection will be its angle with that projection plane.OXYZHVa AaaxbBbbxOXYZHVaAaWabBbbabAH H 投影投影立立標(biāo)標(biāo)差差Z實(shí)實(shí)長(zhǎng)長(zhǎng)True length直角三角形法直角三角形法Right Triangle Methoda ab bA AH H投影投影立立標(biāo)標(biāo)差差ZZ實(shí)長(zhǎng)實(shí)長(zhǎng)True True lengthlengtha a b b A AV V投影投影遠(yuǎn)遠(yuǎn)標(biāo)標(biāo)差差YY實(shí)長(zhǎng)實(shí)長(zhǎng)True True lengthlength a a b b A AW W投影投影橫橫標(biāo)標(biāo)差差XX實(shí)長(zhǎng)實(shí)長(zhǎng)True True lengthlength 斜邊斜邊（Bevel edgeBevel edge ）-實(shí)長(zhǎng)實(shí)長(zhǎng)（True lengthTrue length）斜邊與直角邊夾角斜邊與直角邊夾角 -傾角傾角Angle between the bevel edge and right-angle edge(Angle)Angle between the bevel edge and right-angle edge(Angle)規(guī)律規(guī)律（RuleRule）：兩直角邊兩直角邊（two right-angle edgestwo right-angle edges）投影投影(Projection)(Projection)坐標(biāo)差坐標(biāo)差 (Coordinate difference)(Coordinate difference)直線與投影軸的夾角(Angle between line and projection axis)OXYZHVa AaWa b Bbb abAH H投影投影立標(biāo)差Z實(shí)長(zhǎng)實(shí)長(zhǎng)True lengthZZa ab bA AH H投影投影立立標(biāo)標(biāo)差差ZZ實(shí)長(zhǎng)實(shí)長(zhǎng)True True lengthlengtha a b b A AV V投影投影遠(yuǎn)遠(yuǎn)標(biāo)標(biāo)差差YY實(shí)長(zhǎng)實(shí)長(zhǎng)True True lengthlength a a b b A AW W投影投影橫橫標(biāo)標(biāo)差差XX實(shí)長(zhǎng)實(shí)長(zhǎng)True True lengthlength 斜邊斜邊（Bevel edgeBevel edge ）-實(shí)長(zhǎng)實(shí)長(zhǎng)（True lengthTrue length）斜邊與直角邊夾角斜邊與直角邊夾角 -傾角傾角Angle between the bevel edge and right-angle edgeAngle between the bevel edge and right-angle edge-Angle between line and plane)Angle between line and plane)規(guī)律規(guī)律（RuleRule）：兩直角邊兩直角邊（two right-angle edgestwo right-angle edges）投影投影(Projection)(Projection)坐標(biāo)差坐標(biāo)差 (Coordinate difference)(Coordinate difference)斜邊與另一直角邊夾角斜邊與另一直角邊夾角 -直線與投影軸夾角直線與投影軸夾角Angle between the bevel edge and the other right-angle edge Angle between the bevel edge and the other right-angle edge-Angle between Angle between line and axisline and axisZYX作圖舉例(Problems)例1.已知直線段AB與H面的傾角=30,其他條件如圖,完成AB的另一投影。Complete the other projection of line AB with =30abab=30 ZB1多解題只求一解例例2.2.已知直線段已知直線段ABAB與與H H面的傾角面的傾角=30=30,其他條件其他條件如圖如圖,完成完成ABAB的另一投影。的另一投影。Complete the other projection of line AB with Complete the other projection of line AB with =30=30a ab ba ab b=30=30 Z ZA A1 1結(jié)論結(jié)論:錯(cuò)誤錯(cuò)誤!例2.已知直線段AB與H面的傾角=30,其他條件如圖,完成AB的另一投影。abab=30 ZA1H投影實(shí)長(zhǎng)Z分析:(1)已知(2)又知V投影,可得Z(3)根據(jù)“一角一對(duì)邊”可 作出三角形另一直角 邊-H投影空間兩直線的相對(duì)位置分為：空間兩直線的相對(duì)位置分為：平行平行Parallel、相交相交intersecting、交叉交叉skew。兩直線平行兩直線平行Parallel lines投影特性投影特性：Projection characteristics 空間兩直線平行，空間兩直線平行，則其各則其各同名投影同名投影必相互必相互平行，反之亦然。平行，反之亦然。If two lines in space are parallel,their projections on any projection plane will be parallel,and vice versa.aVHc bcdABCDb d a 四、空間兩直線的相對(duì)位置四、空間兩直線的相對(duì)位置Relative position of two linesabcdc a b d 例例1：判斷圖中兩條直線是否平行。：判斷圖中兩條直線是否平行。AB Parallel with CD?對(duì)于一般位置直線，對(duì)于一般位置直線，只要有兩個(gè)同名投影互只要有兩個(gè)同名投影互相平行，空間兩直線就相平行，空間兩直線就平行。平行。Two general lines are parallel in space if their respective projections on two projection planes are parallel.AB/CDb d c a cbadd b a c 對(duì)于特殊位置直線，對(duì)于特殊位置直線，只有兩個(gè)同名投影互相平只有兩個(gè)同名投影互相平行，空間直線不一定平行。行，空間直線不一定平行。For special position lines,they are not necessarily parallel if only two projections appear to be parallel.求出側(cè)面投影后可知：求出側(cè)面投影后可知：AB與與CD不平行。不平行。AB and CD not parallel 例例2：判斷圖中兩條直線是否平行。：判斷圖中兩條直線是否平行。AB Parallel with CD?求出側(cè)面投影求出側(cè)面投影Obtain the profile projection如何判斷？如何判斷？So how to?HVABCDKabcdka b c k d abcdb a c d kk 兩直線相交兩直線相交Intersecting linesIntersecting lines判別方法：判別方法：若空間兩直線相交，若空間兩直線相交，則其同名投影必相交，則其同名投影必相交，且交點(diǎn)的投影必符合空間一點(diǎn)的投影規(guī)律且交點(diǎn)的投影必符合空間一點(diǎn)的投影規(guī)律。If two lines intersect in space,their corresponding projections also intersect.交點(diǎn)是兩直交點(diǎn)是兩直線的共有點(diǎn)線的共有點(diǎn)(intersecting point is the one which two lines share)cabb a c d k kd例：過(guò)例：過(guò)C點(diǎn)點(diǎn)作水平線作水平線CD與與AB相交。相交。Construct a horizontal line CD intersecting with AB.先作正面投影先作正面投影V projection firstd b a abcdc 1(2 )3(4)兩直線交叉兩直線交叉 Skew linesSkew lines投影特性投影特性 Projection characteristics 同名投影可能相交，但同名投影可能相交，但“交交點(diǎn)點(diǎn)”不符合空間一個(gè)點(diǎn)的投影規(guī)不符合空間一個(gè)點(diǎn)的投影規(guī)律律。Corresponding projections may be intersecting,but“intersecting points”dont abide by the projection rule of one spatial point.“交點(diǎn)交點(diǎn)”是兩直線上的一是兩直線上的一 對(duì)對(duì)重重影點(diǎn)的投影影點(diǎn)的投影，用其可幫助判斷兩，用其可幫助判斷兩直線的空間位置。直線的空間位置。“intersecting points”are the projections of the two overlapped projection points,which can be used for judging the spatial positions of two lines.、是面的重影點(diǎn)是面的重影點(diǎn)(Overlapped projection points)、是是H面的重影點(diǎn)面的重影點(diǎn)為什么？為什么？Why?123 4 兩直線相交嗎？?jī)芍本€相交嗎？Intersecting lines?直線的垂直直線的垂直 PERPENDICULARITY:LINES五、兩直線垂直相交五、兩直線垂直相交(或交叉）或交叉）Perpendicularity of linesPerpendicularity of lines直角邊投影的規(guī)律：直角邊投影的規(guī)律：The theorem of perpendicularity 若直角有一邊平行于投影面，則它在若直角有一邊平行于投影面，則它在該該投影面上的投影仍為直角。投影面上的投影仍為直角。If one of two perpendicular lines is parallel to a projection plane,the projections of both lines on that plane will be at 90 to each other.設(shè)設(shè) 直角邊直角邊BC/H面面因因 BCAB,同時(shí)同時(shí)BCBb所以所以 BCABba平面平面直線在直線在H面上的面上的投影互相垂直投影互相垂直H projections are perpendicular即即 abc為直角為直角因此因此 bcab故故 bc ABba平面平面又因又因 BCbca c b abc.證明：證明：ABCabcH直角投影定理OXYZHVa ab bcc If：ABAC AC/H面Then：acab 即bac=90若投影中有一個(gè)投影互相垂直，且其中一條為該面的平行線，則兩直線在空間也垂直。若直角的任意一條邊平行于某一投影面,則它在該投影面上的投影仍為直角。逆定理If：abac AC/H面Then：ABAC a a ab bcc 例例1:1:判斷下列直線是否垂直判斷下列直線是否垂直Perpendicular lines?Perpendicular lines?垂直Yes!a abbcc 垂直(交叉)Yes!(Skew)kkka a ab bcc 不垂直No!因?yàn)橹本€AB和AC都沒(méi)有平行于正立投影面。The line AC is not perpendicular to line AB,since neither line AB nor AC is parallel to the V plane or H plane.d abca b c d例例2：過(guò)：過(guò)C點(diǎn)作直線與點(diǎn)作直線與AB垂直相交。垂直相交。Construct line perpendicular to line AB through point C AB為正平線為正平線,正正面投影反映直角。面投影反映直角。As a frontal line,the V projection of AB appear the true 90.應(yīng)用舉例應(yīng)用舉例 Application examplesApplication examplesk k Y Yk k a a a abbb bc cc c 分析：過(guò)分析：過(guò)A A作作BCBC的垂線的垂線,垂足為垂足為K K注意注意:求距離包括求距離包括投影投影和和實(shí)長(zhǎng)實(shí)長(zhǎng)DIstanceDIstance means both projections and true length means both projections and true length(1)(1)求點(diǎn)到直線的距離求點(diǎn)到直線的距離 Distance from point to lineDistance from point to line例例3.3.已知已知A A點(diǎn)和直線點(diǎn)和直線BCBC的的V V投影投影和和H H投影投影,求求A A點(diǎn)到點(diǎn)到BCBC的距離。的距離。Find the distance between point A and line BC.Find the distance between point A and line BC.Y Y實(shí)長(zhǎng)實(shí)長(zhǎng) 3 3D:AKD:AK BC BCBC BC/H/H面面 2D:ak 2D:ak bcbcA AB BC CK K步驟步驟StepsSteps:(1)H(1)H：過(guò)：過(guò)a a作作bcbc的垂線的垂線,垂足為垂足為k kH:Draw line perpendicular to bc through H:Draw line perpendicular to bc through point point a a,perpendicular foot,perpendicular foot k k(2)(2)根據(jù)直線投影規(guī)律根據(jù)直線投影規(guī)律,可得可得a a k k ak-a k(3)(3)由直角三角形法可求由直角三角形法可求AKAK實(shí)長(zhǎng)實(shí)長(zhǎng)True length by right Triangle MethodTrue length by right Triangle Methodk kk k a a a ab b b bc c c c 分析分析Analysis Analysis：實(shí)質(zhì)為求交叉兩直線的公垂線問(wèn)題實(shí)質(zhì)為求交叉兩直線的公垂線問(wèn)題Common perpendicularCommon perpendicular(1)(1)公垂線既垂直于直線公垂線既垂直于直線L,L,也垂直于也垂直于直線直線ABAB。Line perpendicular to both AB and L.(2)(2)因?yàn)橐驗(yàn)锳BAB為正垂線為正垂線,所以其公垂線所以其公垂線必平行于必平行于V V面。面。The common perpendicular must be parallel to V because line AB is a V-perpendicular line注意注意:求距離包括求距離包括投影投影和和實(shí)長(zhǎng)實(shí)長(zhǎng)DIstanceDIstance means both projections and true length means both projections and true length(1)(1)求兩交叉直線間的距離求兩交叉直線間的距離Distance between two skew linesDistance between two skew lines例例5.5.已知已知ABAB為正垂線為正垂線,又知直線又知直線L L的的V V投投影和影和H H投影投影,求兩直線間的距離。求兩直線間的距離。AB is a V-perpendicular line.Determine the AB is a V-perpendicular line.Determine the distance between AB and L.distance between AB and L.實(shí)長(zhǎng)實(shí)長(zhǎng)步驟步驟StepsSteps:(1)V(1)V：過(guò)：過(guò)a a 作作ll 的垂線的垂線,垂足為垂足為k k，k k c c 為公垂線的為公垂線的V V投影投影 V:Draw V:Draw k c l(2)(2)根據(jù)點(diǎn)的投影規(guī)律根據(jù)點(diǎn)的投影規(guī)律,可得可得k k K K-K-K(3)(3)由直角投影定理，可得由直角投影定理，可得kckc Right triangle-kc Right triangle-kc (3D:KC (3D:KC AB KC AB KC/V V面面 2 2D:kcD:kc abab）ll ll小結(jié)小結(jié) SummarySummary 一般位置直線一般位置直線General position linesGeneral position lines三個(gè)投影與各投影軸都傾斜。三個(gè)投影與各投影軸都傾斜。All three projections inclined to the axes.All three projections inclined to the axes.投影面平行線投影面平行線lines parallel to projection planeslines parallel to projection planes 在其平行的投影面上的投影反映線段實(shí)長(zhǎng)及與相應(yīng)投影面的在其平行的投影面上的投影反映線段實(shí)長(zhǎng)及與相應(yīng)投影面的夾角。另兩個(gè)投影平行于相應(yīng)的投影軸。夾角。另兩個(gè)投影平行于相應(yīng)的投影軸。When a line is parallel to a projection plane,its projection upon that plane will be the true length of the line.Its projection on the adjacent planes will be parallel to the projection axis.投影面垂直線投影面垂直線Lines perpendicular to a projection plane 在其垂直的投影面上的投影積聚為一點(diǎn)。另兩個(gè)投影反映實(shí)在其垂直的投影面上的投影積聚為一點(diǎn)。另兩個(gè)投影反映實(shí)長(zhǎng)且垂直于相應(yīng)的投影軸。長(zhǎng)且垂直于相應(yīng)的投影軸。When a line is perpendicular to a projection plane,it appears as a point upon that plane.Its projection on the adjacent planes will be perpendicular to the corresponding projection axis and will represent the true length.直角三角形法直角三角形法求實(shí)長(zhǎng)與傾角求實(shí)長(zhǎng)與傾角True length and angles by True length and angles by right triangle methodright triangle method直角投影定理直角投影定理The theorem of perpendicularity 若直角的任意一條邊平行于若直角的任意一條邊平行于某某一投影面一投影面,則它在則它在該該投影面上的投影仍為直角。投影面上的投影仍為直角。If one of two perpendicular lines is parallel to one projection plane,the projections of both lines on that plane will be at 90 to each other.作業(yè)作業(yè)（HomeworkHomework）：P11P11，P12P12，P13P13 ，P14P14 (注意線型、多解題只求一解注意線型、多解題只求一解)參考：教材第參考：教材第 119-131 119-131 頁(yè)頁(yè)Refer to textbook:P119-131Refer to textbook:P119-131下一講：平面的投影下一講：平面的投影(教材第教材第132-142)132-142)next lecture:projection of plane:next lecture:projection of plane:P132-142 P132-142