2013秋季第8讲换面.ppt

问题：如何求异面直线的公垂线？ Problem：How to construct common perpendicular to skew lines?,如果其中一条直线是投射线？ What if one of the skew lines perpendicular to projection plane?,O,X,Y,Z,H,V,a,A,a,C,W,a,b,B,b,c,c,c,b,问题：如何求一般位置平面真形？ Problem： How to get the true shape of an inclined plane?,第八讲 投影变换Lesson 8: Projection transformation,1、问题的提出 Problems,如何求倾斜平面的真实形状？ How to get the true size of an inclined plane?,换 面 法Projection transformation： 物体本身在空间的位置不动，而用某一新投影面（辅助投影面）代替原有投影面，使物体相对新的投影面处于解题所需要的有利位置，然后将物体向新投影面进行投射。With the object staying where it is, using a new projection plane (auxiliary projection plane) to replace the old projection plane such that the problem can be solved by the new projection of the object,解决方法：更换投影面 Solutions: Auxiliary projection plane,2、新投影面的选择原则 Auxiliary projection plane setting,1. 新投影面必须对空间物体处于最有利的解题位置。 The new projection plane must be placed purposely in order to solve the problem,平行于新的投影面 Parallel to the new projection plane 垂直于新的投影面 Perpendicular to the new projection plane,2. 新投影面必须垂直于某一保留的原投影面， 以构成一个相互垂直的两投影面的新体系。 The new projection plane must be perpendicular to one old projection plane which is kept so that they can constitute a new perpendicular projection system.,(1) 更换一次投影面 First projection transformation,A点的两个投影：a， a Two projections of point A: a， a,A点的两个投影：a，a1 Two projections of point A: a， a1,(i) 新投影体系的建立 The construction of new projection system,3、点的投影变换规律 The rule of point projection transformation,ax1,V,H,X,P1,H,X1,a,a,a1,V,H,A,a,ax,X,a1,ax1,(ii) 新旧投影之间的关系 The relationship of old and new projections, aa1 X1, a1ax1 = aax, 点的新投影到新投影轴的距离等于被代替的投影 到原投影轴的距离。,ax,a,一般规律：, 点的新投影和与它有关的原投影的连线，必垂直 于新投影轴。,.,更换H面 replacing H,(iii) 求新投影的作图方法 Drawing of new projection,V,H,X,由点的不变投影向新投影轴作垂线，并在垂线上量取一段距离，使这段距离等于被代替的投影到原投影轴的距离。 Draw line perpendicular to new axis from the unchanged projection, extend the line same distance as from the replaced projection to the old axis,a,a,ax,ax1,ax1,更换V面 replacing V,作图规律：,4、直线的换面 Projection transformation of line,(1) 把一般位置直线变换成投影面平行线 Turning oblique line into line parallel to a projection plane,用P1面代替V面，在P1/H投影体系中，AB/P1。 Replacing V with P1, AB/P1,空间分析：,换H面行吗？ Does replacing H work?,不行！ No!,作图：,新投影轴的位置？ The position of new axis?,与ab平行Parallel to ab,(2) 把面平行线变换成投射线 Turning parallel line into perpendicular to a projection plane,可以通过投影面变换换将一般位置直线转化为面平行线，将面平行线转换为投射线. Now , oblique line can be turned into parallel line and parallel line can be turned into perpendicular to projection plane. 问：是否可将一般位置直线转换为投射线？ Question: Can oblique line be turned into perpendicular to projection plane?,答:用一次换面不可以.(为什么?) Answer: Only one transformation can not make it. (Why?) 可用二次换面实现. With second auxiliary projection plane,二次换面 Secondary auxiliary view,(3)更换两次投影面 Second auxiliary projection plane,(i) 新投影体系的建立 Construction of new projection system,按次序更换 Transforming in sequence,A,a,V,H,a,ax,X,a,a,X,V,H,(ii) 求新投影的作图方法 Steps of drawing,作图规律 a2a1 X2 轴 a2ax2 = aax1,ax,(1) 把一般位置平面变换成投影面垂直面 Transforming oblique plane into plane perpendicular to projection plane,如果把平面内的一条直线变换成新投影面的垂直线，那么该平面则变换成新投影面的垂直面。 If a line in one plane can be transformed into a line perpendicular to new projection plane, that plane then has been transformed into plane perpendicular to new projection plane.,空间分析Spatial analysis：,两平面垂直需满足什么条件？ When are two planes perpendicular to each other?,5、平面的换面 Projection transformation of plane,一般位置直线变换成投影面垂直线，需经几次变换？How many transformations should be needed to turn one oblique line into one perpendicular to projection plane?,在平面内取一条投影面平行线，经一次换面后变换成新投影面的垂直线，则该平面变成新投影面的垂直面。 Construct one horizontal line in the plane and turn it into one perpendicular to the new Vertical plane after one transformation, and the plane be perpendicular to the new Vertical plane.,能否只进行一次变换？Could it be done by only one transformation?,思考： 若变换H面，需在面内取什么位置直线？ What kind of line should be drawn in the plane if one new Horizontal plane needed?,正平线！Frontal line!,a,b,c,a,c,b,X,V,H,例：把三角形ABC变换成投影面垂直面。 Turn the triangle ABC into plane perpendicular to projection plane, 在平面内取一条水平 线AD。 Draw a horizontal line AD in ABC, 将AD变换成新投影 面的垂直线。 Turn AD into line perpendicular to new projection plane,反映平面对哪个投影面的夹角？ is the angle between ABC and which plane?,（2）将投射面换为面平行面 Turn perpendicular plane into parallel to projection plane,能否将一般位置平面换为面平行面？ Can we turn an oblique plane into one parallel to a projection plane?,b,a,c,a,b,c,b1,c1,a1,ABC真形,例：把三角形ABC变换成投影面平行面。 Turning Triangle ABC into plane parallel to a projection plane,X,V,H,一次换面, 把一般位置平面变换成新投影面的垂直面； First transformation of projection plane turns the oblique plane into perpendicular one. 二次换面，再变换成新投影面的平行面。 Second transformation of projection plane turns the perpendicular plane into parallel one.,（3） 把一般位置平面变换成投影面平行面 Turn an oblique plane into one parallel to projection plane,a,b,a,c,b,X,V,H,c,作图Procedure：,AB是水平线 ABH,X2轴的位置？ Position of axis X2?,平面的实形 True shape of the plane,与其平行 Parallel to this,例1 求角平分线的投影 Ex3: Finding the bisector of an angle(ABC),6、换面法的应用 Application of transformation of projection plane,ABD CBD 但（but) abdcbd,例1 求角平分线的投影 Ex1: Finding the projections of the bisector of an angle(ABC),作为特殊情况，当分角线平行于投影面时，这条分角线的投影也就是该角投影的分角线。 As a special case, when the bisector of an angle is parallel to a projection plane, the projection of this bisector will also be the bisector of the projection of the angle.,水平线 Horizontal line,距离Distance,d1,如下图：当直线AB垂直于投影面时，CD平行于投影面，其投影反映实长。As the figure below shows: If ABP, then CDP , and its projection of P is the true length.,求C点到直线AB的距离，就是求垂线CD的实长。The required distance is the true length of perpendicular line CD,过作线平行于x2轴。 Draw line parallel to x2 through c1,如何确定d1点的位置？How to determine the position of point d1?,例3：求两异面直线的距离。,a,c,b,a,b,c,d,d,O,V H,H X1 P1,a1,b1,c1,d1,P1 P2 X2,c2,d2,a2,b2,n2,m2,m1,n1,m2n2 为距离实长。,n1 的位置？ MN为何种位置线？,分析： 只需将其中的一条 直线变换为投影面 垂直线即可。,继续，返回到原投影！,X,MN为P2的平行线！,m1 n1 / O2X2轴。,O1,O2,b,a,a,b,c,d,例4：已知两交叉直线AB和CD的公垂线的长度为MN，且AB为水平线，求CD及MN的投影。 Given: The length of common perpendicular of skew lines AB and CD is MN, ABH Reqd: Complete the projections of CD and MN,V,H,X,圆半径=MN The radius of circle =MN,思考题1：用一根水平位置的管子连接空间两管路AB和CD，确定最短的水平连接管的位置及长度,N,C,D,Q,S,H,G,F,L,作水平面 S,作平面P(含AB)平面Q(含CD),得：水平线 EFGH,L 为水平最短距离实长,I,L,MN 为所求,P,M,B,E,K,A,KL 为所求,- 否!,AB、CD有无数条水平连线, 只有一条为题解要求的最短距离,H,PH,PQ,过已知直线作两平行平面 用一水平面截此两平面 截得的两平行线间的距离即为水平最短距离的长度 此距离垂直于此两平行线 水平连线的方向已知，求与其平行且与M&N相交的线即为所求。,AB与CD都平行于投影面时，其投影的夹角才反映真形（60），因此需将AB与C点所确定的平面变换成投影面平行面。The angle is true size (60) when both AB and CD are parallel to projection plane. So turn the plane determined by AB and C into one parallel to projection plane.,例5：过C点作直线CD与AB相交成60角。 Construct line CD, required the angle between CD and AB is 60,a,b,a,c,b,X,V,H,c,几个解？How many solutions?,两个解！Two!,D点的投影如何返回？How to go back the projections of point D?,思考题2：求直线与平面的 夹角的实际大小。 提示：如果ABC是投射面,m,n,n,m,a,c,b,a