1、3.3 垂直问题 Perpendicularity,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。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：,
2、过点作直线与平面垂直。Construct line through a given point perpendicular to a plane.,过点作平面与直线垂直。Construct plane through a given point perpendicular to a line.,一、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
3、plane(all lines in that plane),P,相交主直线,NPL1N(相交 intersecting)L2N(交叉 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
4、 pair of intersecting lines,l1,l1,l2,l2,a,a,n,n,n正平线的V投影n l1,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 is perpendicular to P,例1: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2：过点A作直线垂直于DEF。Construc
5、t line through point A perpendicular to DEF,注意：不是垂足的正面投影 This is not the frontal projection of perpendicular foot.,注意：不是垂足的平面投影This is not the Horizontal projection of perpendicular foot.,求出直线AB对DEF的穿点才是垂足The piercing point of line AB and DEF is the perpendicular foot.,例3 过定点A作直线L的垂面PConstruct plane
6、 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 plane2)平行面间的距离distance between two planes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
7、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,二、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,
8、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 planes are perpendicular,one plane must contain line perpendicular to another the other plane.,过A点作平面Q垂直于平面P。Construct plane Q through point A perpendicular to plane P,A,a,a,p,p,n,n,q,q,例4：过点A作平面垂直于DEF。Construct plane through point A perpendicular to DEF,解法一：S
9、olution method one:,过点A作DEF的垂线。Construct line perpendicular to DEF,过垂线任作一平面即为所求。Construct a plane containing that perpendicular line.,解法二：Solution method two:,在DEF上任取一直线，比如DF。Select a line in DEF arbitrarily such as DF.,过A点作DF的垂面即为所求。Construct plane through point A perpendicular to DF,三、直线与直线垂直 Perp
10、endicularity between lines,(1)几何条件Geometry condition,(2)投影作图Projection,NP,两直线互相垂直,则必有一线在另一直线的垂面上(或平行于另一直线的垂面）It two lines are perpendicular,one must lie in the perpendicular plane of the other one.,过直线L上A点作垂线 Construct line through point A perpendicular to L,L2与L是什么位置关系？The relative position of L2 a
11、nd L?,a,K,Q,P,L1,L2,NPL1 L1P 或 L2/P,a,l,l,p,p,l1,l1,l2,l2,作业Homework：P24P25-7,8,9复习refer to textbook：P152-P157,3.4 综合问题Synthetical Problems,综合作图题是指点、线、面等几何元素及其相互关系同时出现于问题之中，并且对问题的解答，要求满足的条件较多，需要进行综合分析。,常用的求解方法有：,正推法：从正面入手进行分析推理。,反推法：假定问题已解，再反推回去，找出问题的相互联系或条件，从而得到解题的途径。,交轨法：分析满足条件的几何元素各自的空间轨迹，然后求得可以同
12、时满足所求条件的几何元素或关系，从而得到解题的途径。,一、点线面综合作图题,例1：过点A作一条直线，使其与直线BC垂直相交。Construct a line through point A,required to be perpendicular to and intersecting with line BC,过点A作一主直线平面 AEF与直线BC垂直。Construct main lines AE and AF perpendicular to line BC,作出直线BC对平面AEF的穿点K。Find piercing point K of line BC and AEF,直线AK在直线
13、BC的垂面上，故AKBC，又K点在直线BC上，故直线AK为所求。Line AK meets the requirements,例2：已知等腰三角形ABC的底边AB的V投影ab平行于X轴，AC边的V、H投影ac、ac已知，试完成该等腰三角形的H面投影。Given:AB is the bottom edge of isoceles triangle ABC,abX.Reqd:Complete the H projection of ABC,b,分析：假设等腰三角形ABC的底边AB上的高为CD，因为AB为水平线，故cdab，又H为AB的中点，根据定比性可知ad=bd，因此adcbdc。结论：ac=b
14、c,例3：在平面DEF上求一点K与已知三点A、B、C等距，其中A点和C点的Z坐标相等。Find a point K in plane DEF,required to be equidistant from points A,B and C(AC H),q,q,p,例4：已知三条直线CD、EF、GH，要求作一直线AB平行于CD，且与EF、GH相交。Given:Three lines CD,EF and GH.Reqd:Construct a line AB,let ABCD,and intersecting EF and GH.,p,p,例5：已知一直角ABC，其中AB为一直角边，另一直角边AC
15、平行于正垂面P，且C点距离V面20mm，试完成该三角形的V、H投影。Given:Right angle edge AB of ABC,other right angle edge AC is parallel to V-perpendicular plane P,point C is 20mm from V plane.Reqd:complete the V and H projections of ABC,20mm,例6：过已知点A求作一直线AB，使与已知二交叉直线L1、L2相交（P26-1）。Through given point A construct a line AB to be i
16、ntersecting with two skew lines L1 and L2.,a,a,l1,l2,l1,l2,交轨法intersection of tracks：过A点与L1相交的直线的轨迹；The track of lines through point A and intersecting with L12.过A点与L2相交的直线的轨迹;The track of lines through point A and intersecting with L23.两轨迹的交集；The intersection of two tracks,p,p,反推法Reverse inference：
17、假设此直线AB已作出，只限与L1相交，组成平面P即A点与L1组成的平面P。Assuming that line AB intersecting with line L1 had been found,then the plane which AB and L1 define should be the plane which L1 and point A define.AB在P面内，同时又与L2相交，交点必为P面与L2 的公共点穿点。ABP and intersecting with L2 which means the intersection point should be the pie
18、rcing point of L2 and P,例6：过已知点A求作一直线AB，使与已知二交叉直线L1、L2相交（P26-1）。Through given point A construct a line AB to be intersecting with two skew lines L1 and L2.,例7：过已知点A求作一直线AB，使与已知二交叉直线L1、L2垂直（交叉垂直）（P27-8）。Through given point A construct a line AB to be perpendicular to two skew lines L1 and L2,p,p,交轨法？,提示：垂直只是方向问题，与几何元素的具体位置无关。Key:Perpendicularity is a characteristic of direction which has nothing to do with the practical positions of the geometric elements,l1,题目转换为：过A点作P平面的垂线。Problem transformed to:Through point A construct line perpendicular to plane P,作业：P26,27,28复习：P157-P160,
