画法几何课件：第九讲 点线面综合问题

1、第九讲 点线面综合问题 Lesson 9 Comprehensive 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

2、 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：在平面DEF上求一点K与已知三点A、B

3、、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),例3：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。(P26-4) Given: Three lines CD, EF and GH. Reqd: Construct a line AB, let ABCD, and intersecting EF and GH.,例4：已知一直角ABC，其中AB为一直角边，另一直角边AC平行于正垂面P，且C点距离V面2

4、0mm，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,例5：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 wit

5、h 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 L1 2 .过A点与L2相交的直线的轨迹; The track of lines through point A and intersecting with L2 3.两轨迹的交集； The intersection of two tracks,反推法Reverse inference： 假设此直线AB已作出，

6、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 piercing p

7、oint of L2 and P,例5：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6：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,交轨法？,提示： 垂直只是方

8、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,例7：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 p