2013秋季第6讲几何元素的相对位置n (1).ppt

1、第六讲 几何元素的相对位置 Lesson 6: Relative Position of Geometric elements,2,(从属问题） 一、平行问题 线线平行、线面平行、面面平行 二、相交问题 线线相交、线面相交、面面相交 三、垂直问题 线线垂直、线面垂直、面面垂直,几何元素的相对位置,一、平行问题 Parallel,1 直线与平面平行 Line parallel to Plane,L1/L2 L1/P,a,c,b,m,a,b,c,m,例1：过M点作直线MN平行于平面ABC。 Ex1: Construct line MN through M, required MNABC,有无数解

2、Countless solutions,正平线 A frontal line,例2：过M点作直线MN平行于V面和平面ABC。 Ex2: Construct line MN , required MNV plane and MNABC,c,b,a,m,a,b,c,m,唯一解 Exclusive solution,e,e,b,b,a,f,a,d,f,d,例3：判断直线AB是否平行于DEF. Ex3: Is the line AB parallel to DEF?,P,L2,A,l,l,a,a,L1,l2,l2 ,l1,l1,L,例4: 过A点作与直线L平行的平面P Ex4: Construct p

3、lane P through point A, required P parallel to given line L,直线与平面平行 PARALLELISM: LINES AND PLANES,AB CDEF,ab cdef,例5 过A点作平面P/L，且要求P为正垂面。 Ex5: Construct plane P through point A, required P/L and PV,l,l,a,a,例6 AB/P 且AB=45 求AB Ex6: Reqd: Complete projection of AB,p,p,a,a,b,b,由于P的V投影积聚为直线，则a b平行于P，从而作出a

4、 b 。 PV,plane appears as a line on frontal plane, a bP,2 平面与平面平行 Parallel planes, 若一平面上的两相交直线对应平行于另一平面上的两相交直线，则这两平面相互平行。 If a pair of intersecting lines in one plane is parallel to a pair of intersecting lines in a second plane, the planes are parallel., 若两投影面垂直面相互平行，则它们具有积聚性的那组投影必相互平行。 Two perpendi

5、cular planes are parallel if their collecting projections are parallel.,P,L1,A,p,p,a,a,L2,l1,l1,l2,l2,Q,例7：过A点作与平面P平行的平面Q Ex7: Construct Plane Q, required through point A and parallel to plane P,例8：过A作Q/P。 Construct Plane Q, required through point A and parallel to plane P,PH,a,a,p,QH,q,a,c,e,b,b,a,

6、d,d,f,c,f,e,k,h,k,h,O,X,m,m,例9：判断平面ABDC与平面EFHM是否平行,已知ABCDEFMH Ex9: ABCDEFMH, determine whether or not the two planes ABDC and EFHM are parallel to each other.,由于ek不平行于ac,故两平面不平行。,二、相交问题 Intersections,0.直线与直线 Line and Line 直线与平面Line and Plane 平面与平面Plane and Plane,1、直线与平面相交 Intersection of line and pl

7、ane,直线与平面相交，其交点(穿点)是直线与平面的共有点，即交点既在直线上又在平面内。 The intersection of point (piercing point) is the single common point of a line and a plane.,要讨论的问题Problems to be solved:, 求直线与平面的交点。Find the piercing point,判别可见性，即判别两者之间的相互遮挡关系。 Determine the visibility,18,l,k,p,p,l,特殊位置的直线与平面相交（积聚性法Collecting method）：,a

8、,b,c,m,n,c,n,b,a,m,例10：求直线MN与平面ABC的交点K并判别可见性。,空间及投影分析,平面ABC是一铅垂面，其水平投影积聚成一条直线，该直线与mn的交点即为K点的水平投影。Plane ABC is a H-perpendicular plane, its H projection appears as a line which intersects mn at k., 求交点 Locating the piercing point of intersection,判别可见性 determine the visibility,由水平投影可知，KN段在平面前，故正面投影上kn

9、为可见。,还可通过重影点判别可见性。 Visibility also can be determined by overlapped projection points.,1(2),交点是可见性的分界点,k,m(n),b,m,n,c,b,a,a,c,空间及投影分析 Analysis,直线MN为铅垂线，其水平投影积聚成一个点，故交点K的水平投影也积聚在该点上。 Line MNH, its H projection is collected to a point to which the piercing point is projected., 求交点(Piercing point), 判别可见

10、性(Visibility),点位于平面上，在前；点位于MN上，在后。故k 2为不可见。 Point I of plane is in front of point II of MN , so k 2 is invisible.,1(2),作图Steps,用面上取点法 Locating a point in a plane,例11：求直线MN与平面ABC的交点K并判别可见性。 Ex11: Piercing point K of line MN and plane ABC, determine the visibility.,2、两平面相交 Intersection of planes,两平面相交

11、其交线为直线，交线是两平面的共有线，同时交线上的点都是两平面的共有点。 The intersection of two planes is a straight line containing all points common to the two planes.,要讨论的问题Problems to be solved：, 求两平面的交线 Find intersection,方法: Methods, 确定两平面的两个共有点。 By finding two common points, 确定一个共有点及交线的方向。 By finding one common point and the dir

12、ection of intersection., 判别两平面之间的相互遮挡关系-可见性。 Determine the Visibilities,22,p,q,q,p,特殊位置的平面与一般位置平面相交（积聚性法 Edge-view method）：Intersection of one special position plane and one general position plane,可通过正面投影直观地进行判别。Determine directly by Frontal projection.,a,b,c,d,e,f,c,f,d,b,e,a,m(n),空间及投影分析Analysis,平

13、面 ABC 与 DEF 都为正垂面，它们的正面投影都积聚成直线。交线必为一条正垂线，只要求得交线上的一个点便可作出交线的投影。 Both ABC and DEF are V-perpendicular planes, their intersections must be a V-perpendicular line. With this given direction, there is only one intersection point to be found., 求交线Intersection, 判别可见性Visibility,作图,从正面投影上可看出，在交线左侧，平面ABC在上，其

14、水平投影可见。 Frontal projection shows that on the left of the intersection, plane ABC is above DEF, so that part of ABC is visible in H projection,能否不用重影点判别？ Can it be determined without overlapped projections points?,例12,交线是可见性的分界线,b,c,f,h,a,e,a,b,c,e,f,h,1(2),空间及投影分析,平面EFH是一水平面，它的正面投影有积聚性。ab与ef的交点m 、 b

15、 c与f h的交点n即为两个共有点的正面投影，故mn即MN的正面投影. Plane EFH is a V-perpendicular plane ,its frontal projection appears as a line which intersects ab and b c at m and n. mn is the frontal projection of intersection line MN, 求交线 Intersection, 判别可见性 Visibility,点在FH上，点在BC上，点在上，点在下，故fh可见，n2不可见。 Point I is on the line

16、FH and point II is on the line BC。Point I is above the point II, so fh is visible while n2 is invisible.,作图（Steps）,例13 Ex13,c,d,e,f,a,b,a,b,c,d,e,f,例14 Ex14,投影分析,N 点的水平投影 n 位于def 的外面，说明点 N 位于DEF 所确定的平面内，但不位于DEF 这个图形内。 所以ABC 和 DEF的交线应为 MK。 Horizontal projection of point N is located outside of the de

17、f which means that point N is in the plane defined by DEF but outside of shape of DEF . Intersection line of ABC andDEF is MK,p,p,a,a,l,l,例15 求平面P与三角形ABC的交线 Ex15: Find the intersection line of plane P and ABC,b,b,c,c,3. 两个一般位置平面相交 Intersection of two oblique planes,辅助平面法 Auxiliary planes method,如果两个

18、一般位置平面的同名投影没有互相重叠的部分，则可用辅助平面法求它们的交线。 Auxiliary planes method can be applied for intersection when two planes have no overlapped areas in corresponding projections,辅助平面法 Auxiliary planes method,P,Q,S2,S1,N,M,L,s1,s2,p,p,q,q,m,m,n,n,l,l,例16 求平面P与平面Q的交线 Ex16： Find the intersection of plane P and plane

19、Q.,辅助平面的选择原则: 面平行面或投射面 Auxiliary planes should be parallel or perpendicular planes.,4、 一般位置的直线与平面相交（辅助平面法） 4、 Intersection of oblique line and plane (Auxiliary plane method), 过直线AB作平面S与平面P相交 Construct plane S intersecting with plane H through line AB, 作两平面的交线MN。 Get the intersection MN of H and S.,

20、求直线AB与交线MN的交点。 Find the intersection point of line AB and MN., 交点K即为直线AB对平面P的穿点。 K is the piercing point of line AB to plane P.,辅助平面一般选特殊位置平面。便于利用积聚性求交线。 Auxiliary plane usually is constructed on special position, so intersection can be determined by its concentration.,步骤Steps： 1）包含L作正垂面Sv Construct

21、 V-perpendicular Rv containing line L,2)求正垂面与P的交线MN Find intersection MN of Rv and P.,3）求MN与L的交点K Find intersection K of MN and L,4）判断可见性 Determine the visibility of L,Sv,l,k,p,p,l,k,m,n,m,n,1,2,3,2 (1),例17：求直线L与平面P的交点 Ex17：Find the intersection of line L and plane P.,Sv,R,P,l1,k1,例 18 在上例中添加一条L1的平行

22、线L2, 则L1L2构成 平面R, 求平面P与R的交线 Ex9： Referring to above example, Adding a line L2 parallel to line L1, Find intersection of P and plane R determined by L1L2,p,p,优点: 比辅助平面法精度高 More accurate than auxiliary plane method,l1,k1,K1,L1,K2,L2,l2,l2,k2,k2,Qv,32,如果两个一般位置平面的同名投影有互相重叠的部分，则可用穿点法求它们的交线。 Piercing poin

23、t method can be applied to find intersection of two planes when their corresponding projections have overlapped areas.,作一个平面内两条直线对另一个平面的穿点，连接两穿点的直线即为这两个平面的交线。 Find two piercing points of two lines in one plane to another plane. Line connecting two points is the intersection of two planes.,M,N,利用穿点法求

24、两平面交线Find the intersection of two planes by piercing point method,参考：教材P143P145 ，P146-P151,34,三、 垂直问题Perpendicularity,1、直线与平面垂直 Perpendicularity of line and plane,2、平面与平面垂直垂直 Perpendicularity of two planes,3、直线与直线垂直 Perpendicularity between lines,如果直线垂直于平面内的任意两条相交直线，则与该平面垂直。反之，如果直线与平面垂直，则与平面内的任意直线垂直

25、。A line is perpendicular to a plane if it is perpendicular to at least two nonparallel lines in a plane. Also, a line perpendicular to all lines in a plane if it is perpendicular to that plane.,要讨论的问题Problems to be solved：, 过点作直线与平面垂直。 Construct line through a given point perpendicular to a plane.,

26、过点作平面与直线垂直。 Construct plane through a given point perpendicular to a line.,1. 直线与平面垂直 Perpendicularity of line and plane,直线与平面垂直Line perpendicular to plane,直线垂直于平面内的一对相交直线（一切直线） Line is perpendicular to two intersecting lines in plane (all lines in that plane),P,相交主直线,NP L1N (相交 intersecting) L2N (交

27、叉 Skew),N-法线 Normal of the plane,性质Characteristics: P面所有直线垂直于N All lines in P are perpendicular to N,过N的所有平面垂直于P All Planes containing N are perpendicular to P,平面的坡角与法线的倾角互余 (p + N=90),主直线平面 Main lines plane,平面用一对相交主直线表示 Plane defined by a pair of intersecting lines,l1,l1,l2,l2,a,a,n,n,n正平线的V投影 n l1

28、,n 水平线的H投影 n l2,平面以其他形式表示 Plane defined by other methods,p,p,l,l,l1,l1,L与P平面不垂直 L is not perpendicular to P,L1与P平面垂直 L 1 is perpendicular to P,例19 :L,L1与平面P垂直吗？ LP? L1 P?,用一对相交主直线来判断 Determine by a pair intersecting main lines,例20：过点A作直线垂直于DEF。 Construct line through point A perpendicular to DEF,求出直

29、线AB对DEF的穿点才是垂足The piercing point of line AB and DEF is the perpendicular foot.,例21 过定点A作直线L的垂面P Construct plane P through given point A perpendicular to line L.,p,p,l,l,L,P,a,a,A,应用: 求距离问题Application : Problems of distance,1) 点到平面的距离 distance between point and plane 2) 平行面间的距离distance between two pl

30、anes 步骤Steps: 1. 过定点作面的垂线 (Construct line through given point perpendicular to plane) 2. 求垂线对面的穿点-积聚性法或辅助平面法 (Find piercing point of perpendicular line and plane -Collecting method or Auxiliary plane method) 3. 求定点到穿点的实长-直角三角形法 (True length of distance between given point and piercing point.-Right triangle method),A,A,K,K,P,P,Q,2. 两平面垂直 Perpendicularity of two planes,(1) 几何条件 Geometry condition,(2) 投影作图 Projection,N,Q1,Q2,Q3,P,两平面互相垂直, 则其中一个平面必经过(或平行于)另一个平面的垂线。If two plane