1、Chapter2 Projection of point、line and plane第二章 點、直線和平面的投影1、One side projection of point點的單面投影點的單面投影AA1aconclusion結論結論：One projection of point cant determine its spatial position（a1）HSection 1.Projetion of a pointSection 1.Projetion of a point點的投影點的投影conclusion結論結論：The two sides of the projection can

2、 uniquely determine the location of the point.點的兩面投影點的兩面投影能唯一能唯一地確定點的空間位置。地確定點的空間位置。VHaaaxXHorizontal plane水平投影面水平投影面Vertical plane正立投影面正立投影面Space point空間點（大寫）空間點（大寫）Aprojection of point A on horizontal plane投影點（小寫）投影點（小寫）A的水平投影的水平投影Projection axis投影軸投影軸OXprojection of point A on vertical plane投影點（小

3、寫）投影點（小寫）A的正面投影的正面投影Projection point A on OX axis投影點（小寫）投影點（小寫）A在在OX的投影的投影OV is fixed不動不動Down 90 degrees下翻下翻90XOaaaxVH1、aaOX；（；（ a and areflect the same X coordinates.a和和a反映同一反映同一X坐標）坐標）Projection law of point:點的投影規律點的投影規律:2、aax = A a ； axa = A aVHXOaaaxVHWaaAaaxayazXOZYconclusion結論結論：The three side

4、s of projection can uniquely determine the location of the point.結論：點的三面投影能結論：點的三面投影能唯一唯一地確定點的空間位置。地確定點的空間位置。Profile plane側立投影面側立投影面projection of point A on profile plane投影點（小寫）投影點（小寫）A的側面投影的側面投影三、 Three sides projection of point點的三面投影Fixed不動不動Turn down 90 degrees下翻下翻90Turn right90 degrees后翻后翻90ZXVH

5、WaaaaxayazOYayYxzyyz y 3、aax= a az （ a and a reflect the same Y coordinates a和和a 反映同一反映同一Y坐標坐標）2、 aa OZ；（a and a reflect the same Z coordinates .a 和和a 反映同一反映同一Z 坐標坐標）1、aa OX； （a and areflect the same X coordinates. a和和a反映同一反映同一X坐標坐標）Projection law of point點的投影規律：xy ZVHWaaAaaxayazOXYDistance from poi

6、nt A to the H plane is equal to the point of the Z coordinates.點到點到H面的距離等于點的面的距離等于點的Z坐標坐標，即即Aa=azO=ZADistance from point A to the V plane is equal to the point of the Y coordinates.點到點到V面的距離等于點的面的距離等于點的Y坐標，即坐標，即Aa=ayO=YADistance from point A to the W plane is equal to the point of the X coordinates.

7、點到點到W面的距離等于點的面的距離等于點的X坐標，即坐標，即Aa=axO=XAXA、YA determine a( XA、 YA確定確定a)；XA、ZA determine a （XA、 ZA確定確定a）；）；YA、ZA determine a （YA、 ZA確確定定a）。）。四、The relationship between the projection of the point and the rectangular coordinates點的投影和直角坐標的關系The relationship between the projection of the point and the re

8、ctangular coordinates：點的投影和直角坐標的關系：點的投影和直角坐標的關系：aaExample 1：例：已知A的兩面投影，求另一面的投影（二補三）。ZVHWaaAaaxayazOXYbbBbbxbybzRelative position：相對于投影面：相對于投影面：Up and down上下上下Left an right左右左右Front and back前后前后up上上down下下left左左right右右front前前back后后If X coordinate is bigger，then it is left（X坐標大，點在左）坐標大，點在左）If X coordin

9、ate is smaller，then it is right（X坐標小，點在右）坐標小，點在右）五、Relative position of two points、 coincident point兩點的相對位置、重影點If Z coordinate is smaller，then it is down（Z坐標小，點在下）坐標小，點在下）If Z coordinate is bigger，then it is up（Z坐標大，點在上）坐標大，點在上）If Y coordinate is bigger，then it is front（Y坐標大，點在前）坐標大，點在前）If Y coordin

10、ate is smaller，then it is front（Y坐標大，點在后）坐標大，點在后）abbbaaUp上上Down下下Up上上Down下下Left左左Right右右Left左左Right右右Front前前Back后后Front前前Back后后Relative position of two pointsZVHWaaAaaxayazOXYbb（b） 1、what is coincident point?Projection of two points coincide on a projection plane指空間兩個點在某一投影面上的投影重合。指空間兩個點在某一投影面上的投影重合

11、。Coincidence on H plane A、B水平重影點水平重影點A、BCoincidence on V plane A、C正面重影點正面重影點A、CCoincidence on W plane A、D側面重影點側面重影點A、DBC（c）cc（d）Ddd2、The feature of coincident point重影點的特點重影點的特點The distance of point A to V plane is equal to the distance of point B to V plane.this is Aa=Bb.Coincident pointaaab（b）b（c）c

12、c（d）dd1.Point A and point B are coincident on H plane,and point A is above point B,so point B is invisible, the representation is brackets b,that is (b).水平重影點水平重影點A 、B ,A在上在上B在下，水平投影在下，水平投影b不可見；不可見；aaab（b）b（c）cc（d）dd2.Point A and point C are coincident on V plane,and point A is in front of point C,s

13、o point C is invisible, the representation is brackets c,that is (c ).正面重影點正面重影點A、CA在前在前C在后，正面投影在后，正面投影c不可見；不可見；3.Point A and point D are coincident on W plane,and point A is on the left of point D,so point D is invisible, representation is brackets d,that is (d ).側面重影點側面重影點A、DA在左在左D在右，側面投影在右，側面投影d不

14、可見。不可見。1.What is projection,and it is feature投影的概念及特性；2. characteristic of parallel projection method平行投影特性；3. Three sides projection of point點的三面投影規律；4. Relative position of two points、 coincident point兩點間的相對位置、重影點。 Difficult knowledge難點知識：難點知識：1. Master the formation and characteristics of the Ort

15、hographic projection method掌握正投影法的形成及特性；2. Master the projection of point and the rules of projection掌握點的投影及其規律。Key knowledge重點知識：Section 2.the projection of line第二節 直線的投影161.The included angle between line AB and the H plane is .與與H面夾角為面夾角為ab=ABcos 3.The included angle between line AB and the W pla

16、ne is .與與w面夾角為面夾角為VHWOXZYABababbaab=ABcos ab=ABcos2、 Three plane projection of line AB are shorter than AB直線的三面投影均小于實長。直線的三面投影均小于實長。1、 Three plane projection of line are oblique to axis直線的三面投影均傾斜于投影軸；直線的三面投影均傾斜于投影軸；4.characteristics of the projection投影特性：投影特性：一、 oblique line( general position line)一

17、般位置直線Turn right 90 degrees向后翻向后翻172.The included angle between line AB and the V plane is .與與V面夾角為面夾角為oblique lineXVHWOYYZababbaaxbyazayaybybxbzCharacteristics of the projection:投影特性：投影特性：1.Three plane projection of line are oblique to axis直線的三面投影均傾斜于投影直線的三面投影均傾斜于投影軸；軸；2.Three plane projection of li

18、ne AB are shorter than AB.直線的三面投影均小于實長。直線的三面投影均小于實長。 OZProjection rules投影規律：投影規律：VHWZXOYABababba OZProjection rules投影規律：投影規律：XVHWOYYZababbaaxbyazayaybybxbz OY Projection rules投影規律：投影規律：VHWXZYOABababbaFixed Backward flip 90 degrees后翻后翻90Down 90 degrees下翻下翻90 OY Projection rules投影規律：投影規律：XVHWOYYZababb

19、a OX Projection rules投影規律：投影規律：ZXVHWOYYababbaH Plumb line鉛垂線鉛垂線W W-perpendicular line側垂線側垂線V V-perpendicular line正垂正垂線線VHWXOZYAB（a）babba OY Projection rules of plumb line:投影規律：投影規律：Turn down 90 degrees OY baba（a）bProjection rules of plumb line:投影規律：投影規律： “/OY. aba（ b ）baProjection rules of V-perpend

20、icular line:正投影規律：正投影規律： /OX. （ b ）aabbaProjection rules of W-perpendicular lineVHWOABababbaDo a right triangle,which concludes the real length and the angle of projection. 在投影圖上用幾何作直角三角形的方在投影圖上用幾何作直角三角形的方法求出線段的實長與傾角的方法。法求出線段的實長與傾角的方法。一、一、 Right triangle method直角三角形法的作圖原理直角三角形法的作圖原理Section 3. The re

21、al length of line and the angle of projection 第三節 線段的實長及其對投影面的傾角Hypotenuse AB斜邊斜邊ABDistance difference| |Za-Zb| |距離差距離差| |Za-Zb| |VHWOABababbaZB1B1BAAngle 夾角夾角right angle sideAB1 is equal to the H plane ab.水水 平投影平投影| |ab| |二、 The method of right triangle直角三角形法的作圖方法HypotenusAB斜邊斜邊ABDistance differenc

22、e| Ya-Yb| |距離差距離差| Ya-Yb| |A1BAAngle 夾角夾角A1B=| |ab | |VHWOABababbaZA1HypotenusAB斜邊斜邊ABDistance difference| |Xa-Xb| |距離差距離差| |Xa-Xb| |A2BAtriangle 夾角夾角A2B=| | ab | |VHWOABababbaZA2Summary總結總結： Triangle 夾角夾角 Hypotenus斜邊斜邊Distance difference距離差距離差Projection投影投影VHWOABababbaaba作圖步驟：作圖步驟：1、利用直角三角、利用直角三角形法

23、求形法求AB直線的直線的水平投影水平投影 ；2、過、過a截取水平投截取水平投影影 長，與過長，與過b的聯的聯系線的交點即為所系線的交點即為所求。（兩個解）求。（兩個解）實長實長ZAB=abbb=ab=abVHWOABababbaKkkkThe feature of projection投影特性：投影特性：2. Fixed ratio characteristics定比性：定比性： AK/BK=ak/bk=ak/bk=ak/bk1. From the property從屬性：從屬性：If point locate at the line,then the projection of point

24、must be on the line projecton.點的投影必在直線的同面投影上；點的投影必在直線的同面投影上；Section 4.point on the line第四節 直線上的點一、point on the line直線上的點abcdabcd c a b C不在線段不在線段AB上上 D不在線段不在線段AB上上 d abTrue or false: C、D on line AB or not.試判斷C、D兩點是否在線段AB上。兩種解法：兩種解法：1、定比法；、定比法；2、利用第三面投影。、利用第三面投影。abkabkTrue or false: K on line AB or no

25、t.判斷點K是否在線段AB上。ak kb= a k kb Known conditions:the projection of AB.Question:find a point S,which is on line AB,and AS=L例 已知線段AB的投影,試定出屬于線段AB的點S的投影，使AS的實長等于已知長度L。abba s；4.Find s。ssYABYAB實長實長投影投影LThree conditions:parallel, intersection, noncoplanar三種情況：平行、相交、異面）三種情況：平行、相交、異面）一、一、parallel with each oth

26、er兩直線平行兩直線平行VHWABCDabdcabdcbacdSection 5.Ralative location of two lines兩直線的相對位置aabbdcdcMethod. 判別方法：判別方法：1. Do the projection of W plane作出作出側面投影；側面投影；2.When AB and CD are in the opposite direction,then they are noncoplanar lines.當兩直線的倒向相反時，為異當兩直線的倒向相反時，為異面直線；面直線；3. When AB and CD are in the same dir

27、ection, and ab cd = abcd ,then they are parallel lines. 若若a b c d a b cd則為異面直線。則為異面直線。當兩直線的倒向相同時，當兩直線的倒向相同時，利用定比性判別：利用定比性判別： 若若a b c d = a b c d 則為平行直線；則為平行直線； 若若a b c d a b cd則為異面直線。則為異面直線。True or false:1、 oblique plane( general position plane)一般位置平面ZVHWXOYababbacccBACSection 6、projection of plane平

28、面對投影面的相對位置平面對投影面的相對位置2、the projection is similar with original graph投影的形狀為空間類似投影的形狀為空間類似形形Projection rules:投影規律投影規律oblique plane一般位置平面一般位置平面1、oblique w i t h t h e projection plane與三個投影面均傾斜；與三個投影面均傾斜；ZXVHWOYYcabbaaxbyazaybaaybybxcccycyczbzcxH :vertical plane鉛垂面鉛垂面V :V-perpendicular plane 正垂面正垂面W : V

29、-perpendicular plane 側垂面側垂面2、the other projections are similar with original graph. 其余投影與空間形狀類似。其余投影與空間形狀類似。1、 Horizontal projection accumulate into a line.水平投影積聚成直水平投影積聚成直線（積聚性）線（積聚性）Projection rules:投影規律：投影規律：VHWXZYOababcaccbBCA2、special position of plane特殊位置平面ZXVHWOYYababbacccThe projection of ve

30、rtical plane2、the other projections are similar with original graph. 其余投影與空間形狀類似。其余投影與空間形狀類似。1、 Horizontal projection accumulate into a line.水平投影積聚成直水平投影積聚成直線（積聚性）線（積聚性）Projection rules:投影規律：投影規律：ZXVHWOYYababbacccThe projection of V-perpendicular plane2、the other projections are similar with origina

31、l graph. 其余投影與空間形狀類似。其余投影與空間形狀類似。1.frontal projection accumulate into a line.水平投影積聚成直水平投影積聚成直線（積聚性）線（積聚性）Projection rules:投影規律：投影規律：ZXVHWOYYababbacccThe projection of W-perpendicular plane2、the other projections are similar with original graph. 其余投影與空間形狀類似。其余投影與空間形狀類似。1.Profile projection accumulate

32、 into a line.水平投影積聚成直水平投影積聚成直線（積聚性）線（積聚性）Projection rules:投影規律：投影規律：VHWXZYOababbacccABCH horizontalplaneV frontalplaneW profileplane2、the other projections are parallel with axises. 其余投影平行于投影軸。（其余投影平行于投影軸。（ Z coordinate is equality反映同一反映同一Z坐標）坐標）1、 H projection reflects the real shape水平投影反映實形；水平投影反映

33、實形； （顯實性）（顯實性）Projection rules:Classification:（2）parallel plane投影面的平行面Projection rules.投影規律：投影規律：TheprojectionofhorizontalplaneZXVHWOYYcabbaaxbyazaybaaybybxcccycycxczbz2、the other projections are lines and parallel with axises.其余投影平行于投影軸（其余投影平行于投影軸（ Z coordinate is equality反映同一反映同一Z坐標）坐標）1、 H projec

34、tion reflects the real shape.水平投影反映實形；水平投影反映實形； （顯實性）（顯實性）ZXVHWOYYababbacccProjection rules.投影規律：投影規律：2、the other projections are lines and parallel with axises.其余投影平行于投影軸（其余投影平行于投影軸（ Z coordinate is equality反映同一反映同一Z坐標）坐標）1、 V projection reflects the real shape.水平投影反映實形；水平投影反映實形； （顯實性）（顯實性）ZXVHWOYYaababbcccProjection r