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1、XXXX秋季第5讲平面机械制图,2.3 平面的投影 Projection of Plane,Z,X,Y,平面图形投影总是为平面图形或直线。 A plane surface always projects as a line or a surface.,XXXX秋季第5讲平面机械制图,一、平面的表示法 Representations of planes,不在同一直线上的三个点 Three points not in a straight line,直线及线外一点 A line and a point not on the line,两平行直线 Two parallel lines,两相交直线Tw
2、o intersecting lines,平面图形 Planar shape,XXXX秋季第5讲平面机械制图,二、各种位置的平面 Positions of Planes,存真性True shape,亲似性Similarity,积聚性Collecting, 平面对一个投影面的投影特性Projection characteristics of planes to a single projection plane,XXXX秋季第5讲平面机械制图, 平面在三投影面体系中的投影特性,平面对于三投影面的位置可分为三类:Positions relative to three principal proje
3、ction planes,投影面垂直面Planes perpendicular to projection plane,投影面平行面Planes parallel to projection plane,一般位置平面General position plane,垂直于某一投影面,倾斜于另两个投影面Perpendicular to a projection plane and inclined to the other two,平行于某一投影面,垂直于另两个投影面Parallel to a projection plane and perpendicular to the other two,与
4、三个投影面都倾斜Inclined to three projection planes,XXXX秋季第5讲平面机械制图,O,X,Y,Z,H,V,a,A,a,W,a,b,B,b,b,V 正垂面,3. 投影面垂直面Planes perpendicular to projection plane,C,c,c,c,H 铅垂面,W 侧垂面,XXXX秋季第5讲平面机械制图,a,b,c,a,c,b,c,b,a,亲似性Similar,亲似性Similar,积聚性collecting,铅垂面H-perpendicular plane,投影特性:,在它垂直的投影面上的投影积聚成直线。该直线与投影轴的夹角反映空间平
5、面与另外两投影面夹角的大小。 It appears as a line in the perpendicular projection plane. The angles between that line and the projection axes represented the angles between the plane and the other two projection planes.,另外两个投影面上的投影有亲似性。The other two projections are similar shapes.,XXXX秋季第5讲平面机械制图,例1. 已知多边形的VH投影,求
6、其W投影。Complete the W projection of polygon with the given VH projections.,a,b,a,(b),c,d,d,(c),e,f,g,h,g(h),f(e),a,b,c,d,e,f,g,h,XXXX秋季第5讲平面机械制图,4. 投影面平行面Planes parallel to projection plane,积聚性collecting,积聚性collecting,存真性True shape,水平面Horizontal plane,投影特性:,(1)一个投影存真 One projection is true-show type.(
7、2)另外两投影积聚-面平行于投影轴的线 The other two projections are line-collecting type. (Plane- line axis)(3)积聚性与存真性同时存在 Both collecting and true-show type,XXXX秋季第5讲平面机械制图,O,X,Y,Z,H,V,a,A,a,C,W,a,b,B,b,c,c,c,b,5.一般位置平面General position plane/Oblique plane,XXXX秋季第5讲平面机械制图,3D图形与2D投影有亲似性2D projections is similar to 3D
8、shape,O,X,Y,Z,H,V,a,a,W,a,b,b,c,c,c,b,一般位置平面投影Projections of General position plane,Y,XXXX秋季第5讲平面机械制图,三、平面上的点和直线Points and lines in planes, 平面上取任意直线 Locate arbitrary line in plane,XXXX秋季第5讲平面机械制图,线在面上的条件Line located in the plane if: (1) 过面上的两点 Through two points in the plane (2) 过面上一点, 且平 行于面上另一直线 T
9、hrough one point of the plane and parallel to one line in the plane,XXXX秋季第5讲平面机械制图,a,b,c,b,c,a,d,n,m,例2:已知平面由直线AB、AC所确定,试在平面内任作一条直线。 Construct a line arbitrarily in plane defined by lines AB and AC.,解法一Solution1,解法二Solution2,根据定理二By rule 2,根据定理一By rule 1,XXXX秋季第5讲平面机械制图,例3:在平面ABC内作一条水平线,使其到H面的距 离为1
10、0mm。 Construct a horizontal line 10mm away from H projection plane in plane ABC,n,m,n,m,唯一解!The exclusive solution!,XXXX秋季第5讲平面机械制图,点在面上的几何条件: 必在面内的一条直线上Point lies in a plane if it lies on a line in the plane., 平面上取点 Locating a point in a plane,先找出过此点而又在平面内的一条直线作为辅助线,然后再在该直线上确定点的位置。 A point can be l
11、ocated in a plane by first constructing a auxiliary line on the plane through the point.,面上取点的方法:,面上取点必先取线 Locating a point in plane must be by first locating a line.,XXXX秋季第5讲平面机械制图,步骤Steps:在P面内过A点作直线L Draw line L on P through point A.,例4: A是平面P上一点, 已知A点的V投影, 求H投影AP, find the H projection.,分析: 若A点在
12、平面P上, 必在过A点的直线L上(L在P上) AP A L (L P ),p,p,2,1,1,2,l,l,L,P,a,a,A,(2) 作直线L的H投影Draw H projection of line L.,(3) 根据从属性作出直线L上的A点的H投影Get the H projection of point A.,XXXX秋季第5讲平面机械制图,例5:已知K点在平面ABC上,求K点的水平投影。K plane ABC , find the H projection.,利用平面的积聚性求解By collecting type plane,通过在面内作辅助线求解By auxiliary line,
13、XXXX秋季第5讲平面机械制图,k,b,例6:已知AC为正平线,补全平行四边形ABCD的水平投影。ACV, Complete the H projection of parallelogram ABCD,解法一Solution 1,解法二Solution 2,XXXX秋季第5讲平面机械制图,3.过点或直线作平面Construct plane through point or line,过空间中任意一点A作平面Construct plane through point A,L,过空间中任意一直线L作平面Construct plane through line L,结论: 要有约束条件才能得到唯一
14、解Conclusion: Exclusive solution by extra constraints,A,XXXX秋季第5讲平面机械制图,过直线L作平面Construct plane through line L,l,l,a,a,m,m,l,l,a,a,m,m,过点A作平面Construct plane through point A,思考:能否作出面平行面?,投射面呢?,XXXX秋季第5讲平面机械制图,过直线L可以作什么位置平面?What kind of planes can be constructed through line L?,l,l,水平面( ) 正垂面( ) 正平面( )
15、铅垂面( )侧平面( ) 侧垂面( )一般位置平面( ),XXXX秋季第5讲平面机械制图,过一般位置直线作正垂面 Construct a V-perpendicular plane through a general position line.,l,l,过一般位置直线作铅垂面Construct a V-perpendicular plane through a general position line.,l,l,p,p,p,p,XXXX秋季第5讲平面机械制图,四、平面上的特殊直线 Special lines on plane,O,X,Y,Z,H,V,L1,W,P,1. 主直线 Main l
16、ine,平面上与投影面平行的直线Lines parallel to projection planes,正平线L1 V,水平线L2 H,侧平线L3 W,L2,L3,平行于平面P与投影面的交线Parallel to the intersect of plane P and projection plane,XXXX秋季第5讲平面机械制图,2 主直线平面 Main lines plane,用一对相交主直线表示的平面Plane represented by a pair of intersecting main lines,l1,l1,l2,l2,l1,l1,l2,l2,XXXX秋季第5讲平面机械制
17、图,最大斜度线Maximum inclination line,平面P与H的夹角,称为坡角最大斜度线与H的夹角,3. 最大斜度线Maximum inclination line,XXXX秋季第5讲平面机械制图,Y,定义:平面上与主直线垂直的直线。也称最大坡度线。Line perpendicular to main line on plane.,(2) 特点:最大斜度线的一个投影垂直于主直线。One projection of maximum inclination line perpendicular to main line projection.,最大斜度线的倾角为P平面与该投影面的倾角T
18、he angle of maximum inclination line is the angle between plane and projection plane.,最大斜度线 Maximum inclination line,XXXX秋季第5讲平面机械制图,l,l,a,b,a,b,.,思考:(1)如何求平面P对H面的坡角? Angle between P and H?(2)如何求平面P对V面的坡角? Angle between P and H?,XXXX秋季第5讲平面机械制图,例7、 已知平面P的VH投影, 求平面P对V面的坡角. Find angle between plane P
19、and V.,4. 应用举例 example of application,p ,p,l ,l,实长,分析: 求坡角, 即最大斜度 线对 V面的倾角 is the angle between maximum inclination line and V projection plane.,实长True length,V投影,Y,Y,步骤steps: (1)作平行于V面的主直线LConstruct line LV,(2)作垂直于L的最大斜度线Construct the maximum inclination lineL,(3)用直角三角形法求最大斜度线对V面的倾角Find by right tr
20、iangle method.,思考: 如何 求坡角?,XXXX秋季第5讲平面机械制图,例8 已知正方形的边AB,又知正方形面与H面倾角为45,完成其投影。Square ABCD, = 45,complete projection.,a,a,b,b,2、根据AD为正方形面的最大斜度线,AD的H面倾角为45度。 AD is the maximum inclination line, =45 3、AD的实长为ab, 求出AD的H投影。 True lentgh of AD is ab.,1、由直角投影定理可知两直角边的H投影垂直。AB AD,ab=AD,ad,d,c,d,c,Z,XXXX秋季第5讲平面
21、机械制图,例9、 求作过直线AB的P平面, 要求使P与H面的坡角 =30Construct plane P through line AB, and P =30,b,a,a,实长True length,实长True length,H投影,Z,Z,步骤Steps:,(1)作垂直于AB的最大斜度线BC Find maximum inclination line AB, AB BC,(2)用直角三角形法求最大斜度线对H面的倾角 Find the P by right triangle method.,思考: 如果要求坡角 =30, 能作吗?Can the P with p =30 be constr
22、ucted ?,b,c,c,XXXX秋季第5讲平面机械制图,五、平面图形的投影 Projection of planar shape,1. 空间中平面的投影 (1) 形成特点: 三点共面 Three points in plane (2) 投影特点: 一个平面图形的各个投影之间互为亲似形 Similarity between the three projections2. 平面图形的投影作图 依据: 从属性 Belonging 亲似性 Similar 封闭性 Close,XXXX秋季第5讲平面机械制图,例10. 已知多边形ABCD的abcd投影及ab ,AD是水平线,求其V投影。Given: Projection abcd and ab of Quadrangle ABCD, AD H Reqd: Complete the V projection,a,b,a,b,c,d,c,d,e,e,XXXX秋季第5讲平面机械制图,例11. 已知菱形ABCD的H投影,AB是水平线,求其V投影。Given :Diamond shape ABCD , AB H Reqd :Complete V projection,a,b,a,b,实长True length,c,d,c,d,Z,