第七讲 相交问题 Lesson 7： Intersections直线与直线 Line and Line直线与平面Line and Plane平面与平面Plane and Plane平面与平面立体 Plane and Planar solid曲面与平面立体Surface and Planar solid平面立体与平面立体Planar solid and Planar solid平面立体与曲面立体Planar solid and Surface solid曲面立体与曲面立体Surface solid and Surface solid1、直线与平面相交Intersection of line and plane 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 判别两者之间的相互遮挡关系，即判别可见性。
Determine the visibility特殊位置的直线与平面相交（积聚性法Collecting method）：Intersection of special position line and planeabcmncnbam例1：求直线MN与平面ABC的交点K并判别可见性Ex1: Piercing point K of line MN and plane ABC, determine the visibility.空间及投影分析Analysis 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为可见The H projection shows that KN is in front of plane, so the kn is visibal还可通过重影点判别可见性。
Visibility also can be determined by overlapped projection points.k1(2)k21km(n)bmncbaac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)k21作图Steps用面上取点法Locating a point in a plane例2：求直线MN与平面ABC的交点K并判别可见性Ex2: Piercing point K of line MN and plane ABC, determine the visibility.穿点(特殊位置)PIERCING POINTS (SPECIAL POSITIONS)2、两平面相交 Intersection of planes 两平面相交其交线为直线，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 direction of intersection. 判别两平面之间的相互遮挡关系-可见性 Determine the Visibilities 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.abcdefcfdbeam(n)空间及投影分析Analysis 平面 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 projectionnm能否不用重影点判别？Can it be determined without overlapped projections points?能!Yes!如何判别？How?例3 Ex3 bcfhaeabcefh1(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 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 FH and point II is on the line BCPoint I is above the point II, so fh is visible while n2 is invisible.作图（Steps）mn2nm1例4 Ex4cdefababcdef例5 Ex5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 def which means that point N is in the plane defined by DEF but outside of shape of DEF . Intersection line of ABC andDEF is MKnnmkmkppaall例6 求平面P与三角形ABC的交线Ex6: Find the intersection line of plane P and ABCbbcc3、 两个一般位置平面相交3、Intersection of two oblique planes辅助平面法 Auxiliary planes method 如果两个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 projectionsS1S2PQBA 辅助平面法Auxiliary planes methodPQS2S1NMLs1s2ppmmnnll 例7 求平面P与平面Q的交线Ex7： Find the intersection of plane P and plane 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)PSNMKBABA 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. 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.例8：求直线L与平面P的交点 Ex8：Find the intersection of line L and plane P.步骤Steps：1）包含L作正垂面Rv Construct V-perpendicular Rv containing line L2)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KFind intersection K of MN and L4）判断可见性Determine the visibility of LRvlkpplkmnmn1232 (1)pvRPl1k1例 9 在上例中添加一条L1的平行线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 plan。