2 Projection of point,line and plane 点、直线和平面的投影

1、Chapter2 Projection of point、line and plane第二章 点、直线和平面的投影1、One side projection of point点的单面投影点的单面投影AA1aconclusion结论结论：One projection of point cant determine its spatial position（a1）HSection 1.Projetion of a pointSection 1.Projetion of a point点的投影点的投影conclusion结论结论：The two sides of the projection can

2、 uniquely determine the location of the point.点的两面投影点的两面投影能唯一能唯一地确定点的空间位置。地确定点的空间位置。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空间点（大写）空间点（大写）Aprojection of point A on horizontal plane投影点（小写）投影点（小写）A的水平投影的水平投影Projection axis投影轴投影轴OXprojection of point A on vertical plane投影点（小

3、写）投影点（小写）A的正面投影的正面投影Projection point A on OX axis投影点（小写）投影点（小写）A在在OX的投影的投影OV is fixed不动不动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坐标）坐标）Projection law of point:点的投影规律点的投影规律:2、aax = A a ； axa = A aVHXOaaaxVHWaaAaaxayazXOZYconclusion结论结论：The three side

4、s of projection can uniquely determine the location of the point.结论：点的三面投影能结论：点的三面投影能唯一唯一地确定点的空间位置。地确定点的空间位置。Profile plane侧立投影面侧立投影面projection of point A on profile plane投影点（小写）投影点（小写）A的侧面投影的侧面投影三、 Three sides projection of point点的三面投影Fixed不动不动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坐标坐标）2、 aa OZ；（a and a reflect the same Z coordinates .a 和和a 反映同一反映同一Z 坐标坐标）1、aa OX； （a and areflect the same X coordinates. a和和a反映同一反映同一X坐标坐标）Projection law of point点的投影规律：xy ZVHWaaAaaxayazOXYDistance from poi

6、nt A to the H plane is equal to the point of the Z coordinates.点到点到H面的距离等于点的面的距离等于点的Z坐标坐标，即即Aa=azO=ZADistance from point A to the V plane is equal to the point of the Y coordinates.点到点到V面的距离等于点的面的距离等于点的Y坐标，即坐标，即Aa=ayO=YADistance from point A to the W plane is equal to the point of the X coordinates.

7、点到点到W面的距离等于点的面的距离等于点的X坐标，即坐标，即Aa=axO=XAXA、YA determine a( XA、 YA确定确定a)；XA、ZA determine a （XA、 ZA确定确定a）；）；YA、ZA determine a （YA、 ZA确确定定a）。）。四、The relationship between the projection of the point and the rectangular coordinates点的投影和直角坐标的关系The relationship between the projection of the point and the re

8、ctangular coordinates：点的投影和直角坐标的关系：点的投影和直角坐标的关系：aaExample 1：例：已知A的两面投影，求另一面的投影（二补三）。ZVHWaaAaaxayazOXYbbBbbxbybzRelative position：相对于投影面：相对于投影面：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坐标大，点在左）坐标大，点在左）If X coordin

9、ate is smaller，then it is right（X坐标小，点在右）坐标小，点在右）五、Relative position of two points、 coincident point两点的相对位置、重影点If Z coordinate is smaller，then it is down（Z坐标小，点在下）坐标小，点在下）If Z coordinate is bigger，then it is up（Z坐标大，点在上）坐标大，点在上）If Y coordinate is bigger，then it is front（Y坐标大，点在前）坐标大，点在前）If Y coordin

10、ate is smaller，then it is front（Y坐标大，点在后）坐标大，点在后）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指空间两个点在某一投影面上的投影重合。指空间两个点在某一投影面上的投影重合

11、。Coincidence on H plane A、B水平重影点水平重影点A、BCoincidence on V plane A、C正面重影点正面重影点A、CCoincidence on W plane A、D侧面重影点侧面重影点A、DBC（c）cc（d）Ddd2、The feature of coincident point重影点的特点重影点的特点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).水平重影点水平重影点A 、B ,A在上在上B在下，水平投影在下，水平投影b不可见；不可见；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 ).正面重影点正面重影点A、CA在前在前C在后，正面投影在后，正面投影c不可见；不可见；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 ).侧面重影点侧面重影点A、DA在左在左D在右，侧面投影在右，侧面投影d不

14、可见。不可见。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点的三面投影规律；4. Relative position of two points、 coincident point两点间的相对位置、重影点。 Difficult knowledge难点知识：难点知识：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掌握点的投影及其规律。Key knowledge重点知识：Section 2.the projection of line第二节 直线的投影161.The included angle between line AB and the H plane is .与与H面夹角为面夹角为ab=ABcos 3.The included angle between line AB and the W pla

16、ne is .与与w面夹角为面夹角为VHWOXZYABababbaab=ABcos ab=ABcos2、 Three plane projection of line AB are shorter than AB直线的三面投影均小于实长。直线的三面投影均小于实长。1、 Three plane projection of line are oblique to axis直线的三面投影均倾斜于投影轴；直线的三面投影均倾斜于投影轴；4.characteristics of the projection投影特性：投影特性：一、 oblique line( general position line)一

17、般位置直线Turn right 90 degrees向后翻向后翻172.The included angle between line AB and the V plane is .与与V面夹角为面夹角为oblique lineXVHWOYYZababbaaxbyazayaybybxbzCharacteristics of the projection:投影特性：投影特性：1.Three plane projection of line are oblique to axis直线的三面投影均倾斜于投影直线的三面投影均倾斜于投影轴；轴；2.Three plane projection of li

18、ne AB are shorter than AB.直线的三面投影均小于实长。直线的三面投影均小于实长。 OZProjection rules投影规律：投影规律：VHWZXOYABababba OZProjection rules投影规律：投影规律：XVHWOYYZababbaaxbyazayaybybxbz OY Projection rules投影规律：投影规律：VHWXZYOABababbaFixed Backward flip 90 degrees后翻后翻90Down 90 degrees下翻下翻90 OY Projection rules投影规律：投影规律：XVHWOYYZababb

19、a OX Projection rules投影规律：投影规律：ZXVHWOYYababbaH Plumb line铅垂线铅垂线W W-perpendicular line侧垂线侧垂线V V-perpendicular line正垂正垂线线VHWXOZYAB（a）babba OY Projection rules of plumb line:投影规律：投影规律：Turn down 90 degrees OY baba（a）bProjection rules of plumb line:投影规律：投影规律： “/OY. aba（ b ）baProjection rules of V-perpend

20、icular line:正投影规律：正投影规律： /OX. （ b ）aabbaProjection rules of W-perpendicular lineVHWOABababbaDo a right triangle,which concludes the real length and the angle of projection. 在投影图上用几何作直角三角形的方在投影图上用几何作直角三角形的方法求出线段的实长与倾角的方法。法求出线段的实长与倾角的方法。一、一、 Right triangle method直角三角形法的作图原理直角三角形法的作图原理Section 3. The re

21、al length of line and the angle of projection 第三节 线段的实长及其对投影面的倾角Hypotenuse AB斜边斜边ABDistance difference| |Za-Zb| |距离差距离差| |Za-Zb| |VHWOABababbaZB1B1BAAngle 夹角夹角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直角三角形法的作图方法HypotenusAB斜边斜边ABDistance differenc

22、e| Ya-Yb| |距离差距离差| Ya-Yb| |A1BAAngle 夹角夹角A1B=| |ab | |VHWOABababbaZA1HypotenusAB斜边斜边ABDistance difference| |Xa-Xb| |距离差距离差| |Xa-Xb| |A2BAtriangle 夹角夹角A2B=| | ab | |VHWOABababbaZA2Summary总结总结： Triangle 夹角夹角 Hypotenus斜边斜边Distance difference距离差距离差Projection投影投影VHWOABababbaaba作图步骤：作图步骤：1、利用直角三角、利用直角三角形法

23、求形法求AB直线的直线的水平投影水平投影 ；2、过、过a截取水平投截取水平投影影 长，与过长，与过b的联的联系线的交点即为所系线的交点即为所求。（两个解）求。（两个解）实长实长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从属性：从属性：If point locate at the line,then the projection of point

24、must be on the line projecton.点的投影必在直线的同面投影上；点的投影必在直线的同面投影上；Section 4.point on the line第四节 直线上的点一、point on the line直线上的点abcdabcd c a b C不在线段不在线段AB上上 D不在线段不在线段AB上上 d abTrue or false: C、D on line AB or not.试判断C、D两点是否在线段AB上。两种解法：两种解法：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.判断点K是否在线段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 已知线段AB的投影,试定出属于线段AB的点S的投影，使AS的实长等于已知长度L。abba s；4.Find s。ssYABYAB实长实长投影投影LThree conditions:parallel, intersection, noncoplanar三种情况：平行、相交、异面）三种情况：平行、相交、异面）一、一、parallel with each oth

26、er两直线平行两直线平行VHWABCDabdcabdcbacdSection 5.Ralative location of two lines两直线的相对位置aabbdcdcMethod. 判别方法：判别方法：1. Do the projection of W plane作出作出侧面投影；侧面投影；2.When AB and CD are in the opposite direction,then they are noncoplanar lines.当两直线的倒向相反时，为异当两直线的倒向相反时，为异面直线；面直线；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则为异面直线。则为异面直线。当两直线的倒向相同时，当两直线的倒向相同时，利用定比性判别：利用定比性判别： 若若a b c d = a b c d 则为平行直线；则为平行直线； 若若a b c d a b cd则为异面直线。则为异面直线。True or false:1、 oblique plane( general position plane)一般位置平面ZVHWXOYababbacccBACSection 6、projection of plane平

28、面对投影面的相对位置平面对投影面的相对位置2、the projection is similar with original graph投影的形状为空间类似投影的形状为空间类似形形Projection rules:投影规律投影规律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与三个投影面均倾斜；与三个投影面均倾斜；ZXVHWOYYcabbaaxbyazaybaaybybxcccycyczbzcxH :vertical plane铅垂面铅垂面V :V-perpendicular plane 正垂面正垂面W : V

29、-perpendicular plane 侧垂面侧垂面2、the other projections are similar with original graph. 其余投影与空间形状类似。其余投影与空间形状类似。1、 Horizontal projection accumulate into a line.水平投影积聚成直水平投影积聚成直线（积聚性）线（积聚性）Projection rules:投影规律：投影规律：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. 其余投影与空间形状类似。其余投影与空间形状类似。1、 Horizontal projection accumulate into a line.水平投影积聚成直水平投影积聚成直线（积聚性）线（积聚性）Projection rules:投影规律：投影规律：ZXVHWOYYababbacccThe projection of V-perpendicular plane2、the other projections are similar with origina

31、l graph. 其余投影与空间形状类似。其余投影与空间形状类似。1.frontal projection accumulate into a line.水平投影积聚成直水平投影积聚成直线（积聚性）线（积聚性）Projection rules:投影规律：投影规律：ZXVHWOYYababbacccThe projection of W-perpendicular plane2、the other projections are similar with original graph. 其余投影与空间形状类似。其余投影与空间形状类似。1.Profile projection accumulate

32、 into a line.水平投影积聚成直水平投影积聚成直线（积聚性）线（积聚性）Projection rules:投影规律：投影规律：VHWXZYOababbacccABCH horizontalplaneV frontalplaneW profileplane2、the other projections are parallel with axises. 其余投影平行于投影轴。（其余投影平行于投影轴。（ Z coordinate is equality反映同一反映同一Z坐标）坐标）1、 H projection reflects the real shape水平投影反映实形；水平投影反映

33、实形； （显实性）（显实性）Projection rules:Classification:（2）parallel plane投影面的平行面Projection rules.投影规律：投影规律：TheprojectionofhorizontalplaneZXVHWOYYcabbaaxbyazaybaaybybxcccycycxczbz2、the other projections are lines and parallel with axises.其余投影平行于投影轴（其余投影平行于投影轴（ Z coordinate is equality反映同一反映同一Z坐标）坐标）1、 H projec

34、tion reflects the real shape.水平投影反映实形；水平投影反映实形； （显实性）（显实性）ZXVHWOYYababbacccProjection rules.投影规律：投影规律：2、the other projections are lines and parallel with axises.其余投影平行于投影轴（其余投影平行于投影轴（ Z coordinate is equality反映同一反映同一Z坐标）坐标）1、 V projection reflects the real shape.水平投影反映实形；水平投影反映实形； （显实性）（显实性）ZXVHWOYYaababbcccProjection r