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单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,2021/9/23,*,*,a,a,a,b,b,b,22 直线的投影,Projection of Line,两点确定一条直线，将两点的同名投影用直线连接，就得到直线的同名投影。,(,Projections of 2 points determine the projection of line.),直线对一个投影面的投影特性,(To single projection plane),一、直线的投影特性,Characteristics,A,B,a,b,直线垂直于投影面,投影重合为一点,积聚性,Point-wise projection,直线平行于投影面,投影反映线段实长,存真性 ab=AB,True length projection,直线倾斜于投影面,投影比空间线段短,ab=ABcos,Foreshorten projection,A,B,a,b,A,M,B,abm,2021/9/23,1,二、直线在三个投影面中的投影特性,Lines projected to the 3-projection-plane,投影面平行线,Lines,parallel to a,Projection plane,平行于某一投影面而,与其余两投影面倾斜,投影面垂直线,Lines perpendicular to a projection plane,正平线,Frontal lines,（,面）,侧平线,Profile lines,（,面）,水平线,Horizontal lines,（,面）,正垂线,（W-perpendicular lines）,（,面）,侧垂线,（W-perpendicular lines）,（,面）,铅垂线,（H-perpendicular lines）,（,面）,一般位置直线,General Position lines,与三个投影面都倾斜的直线,Inclined to all three projection planes,统称特殊位置直线,(Special position lines),垂直于某一,投影面,2021/9/23,2,b,a,a,b,a,b,b,a,a,b,b,a,投影面平行线,Lines parallel to a Projection plane,在其平行的那个投影面上的投影反映实长，并反映直线与另两投影面倾角的实大。,另两个投影面上的投影平行于相应的投影轴。,When a line is parallel to a projection plane,its projection upon that plane will be the true length of the line.Its projection on the adjacent planes will be parallel to the projection axis.,水平线,Horizontal line,侧平线,Profile line,正平线,Frontal line,投 影 特 性：,与H面的夹角,（,Angle with H,）,:,与V面的角:与W面的夹角:,实长,（True length）,实长,实长,b,a,a,a,b,b,2021/9/23,3,投影面垂直线,Lines perpendicular to a projection plane,铅垂线,H-perpendicular line,正垂线,V-perpendicular line,侧垂线,W-perpendicular line,另外两个投反映线段实长。且垂直于相应的投影轴。,在其垂直的投影面上，,投影有积聚性,。,投影特性:,c,(,d,),c,d,d,c,a,b,a(b),a,b,e,f,e,f,e,(,f,),When a line is perpendicular to a projection plane,it appears as a point upon that plane.Its projection on the adjacent planes will be perpendicular to the corresponding projection axis and will represent the true length.,2021/9/23,4,一般位置直线,General position lines,投影特性：,三个投影都缩短。即:都不反映空间线段的实长及与三个投影面夹角的实大，且与三根投影轴都倾斜。,All three projections are foreshortened.,That is:Its true length and angles will not be represented in any of these projections.,a,b,b,a,b,a,2021/9/23,5,三、直线上的点,Points on lines,若点在直线上,则点的投影必在直线的同名投影上。,If a point in space lies on a line,the projections of that point also lie on the corresponding projections of the line.,判别方法,：,A,B,C,V,H,b,c,c,b,a,a,2021/9/23,6,点C不,在直线AB上,(No),例1：判断点C是否在线段AB上。,Determine if the point C on line AB,a,b,c,a,b,c,c,a,b,c,a,b,点C在直线AB上,(Yes),2021/9/23,7,例2：判断点K是否在线段AB上。,Determine if the point K on line AB,a,b,k,因k,不在a,b,上，,故点K不在AB上。,k,is not on a,b,then K lies on line AB.,a,b,k,a,b,k,当直线的投影垂直于投影轴时有例外出现，这种情况可增加第三投影。,Exception may occur to special position line.A profile projection is added in this case.,2021/9/23,8,四、,一般位置直线的实长与倾角,True length and angle of a general line,O,X,H,V,a,a,a,x,b,b,b,x,倾角:直线与在某一投影面上的投影间的夹角为直线与该面的倾角,Angle between line and its projection will be its angle with that projection plane.,O,X,Y,Z,H,V,a,A,a,a,x,b,B,b,b,x,2021/9/23,9,O,X,Y,Z,H,V,a,A,a,W,a,b,B,b,b,a,b,A,H 投影,立标差,Z,实长,True length,直角三角形法,Right Triangle Method,2021/9/23,10,a,b,A,H投影,立标差,Z,实长,True length,a,b,A,V投影,远标差,Y,实长,True length,a,b,A,W投影,横标差,X,实长,True length,斜边,（,Bevel edge,）,-实长,（True length）,斜边与直角边夹角,-倾角,Angle between the bevel edge and right-angle edge(Angle),规律,（Rule）,：,两直角边,（two right-angle edges）,投影,(Projection),坐标差,(Coordinate difference),2021/9/23,11,直线与投影轴的夹角,(Angle between line and projection axis),O,X,Y,Z,H,V,a,A,a,W,a,b,B,b,b,a,b,A,H投影,立标差,Z,实长True length,Z,Z,2021/9/23,12,a,b,A,H投影,立标差,Z,实长,True length,a,b,A,V投影,远标差,Y,实长,True length,a,b,A,W投影,横标差,X,实长,True length,斜边,（,Bevel edge,）,-实长,（True length）,斜边与直角边夹角,-倾角,Angle between the bevel edge and right-angle edge,-,Angle between line and plane),规律,（Rule）,：,两直角边,（two right-angle edges）,投影,(Projection),坐标差,(Coordinate difference),斜边与另一直角边夹角,-直线与投影轴夹角,Angle between the bevel edge and the other right-angle edge,-,Angle between line and axis,Z,Y,X,2021/9/23,13,3.作图举例(Problems),例1.已知直线段AB与H面的倾角,=30,其他条件如图,完成AB的另一投影。,Complete the other projection of line AB with,=30,a,b,a,b,=30,Z,B,1,2021/9/23,14,例2.已知直线段AB与H面的倾角,=30,其他条件如图,完成AB的另一投影。,Complete the other projection of line AB with,=30,a,b,a,b,=30,Z,A,1,H投影,b,1,实长,Z,2021/9/23,15,空间两直线的相对位置分为：,平行,Parallel,、,相交,intersecting,、,交叉,skew,。,两直线平行,Parallel lines,投影特性,：,Projection characteristics,空间两直线平行，则其各,同名投影,必相互平行，反之亦然。,If two lines in space are parallel,their projections on any projection plane will be parallel,and vice versa.,a,V,H,c,b,c,d,A,B,C,D,b,d,a,五、空间两直线的相对位置,Relative position of two lines,2021/9/23,16,a,b,c,d,c,a,b,d,例1：判断图中两条直线是否平行。,AB Parallel with CD?,对于一般位置直线，只要有两个同名投影互相平行，空间两直线就平行。,Two,general,lines are parallel in space if their respective projections on,two,projection planes are parallel.,AB/CD,2021/9/23,17,b,d,c,a,c,b,a,d,d,b,a,c,对于特殊位置直线，只有两个同名投影互相平行，空间直线不一定平行。,For special position lines,they are not necessarily parallel if only two projections appear to be parallel.,求出侧面投影后可知：,AB与CD不平行。,AB and CD not parallel,例2：判断图中两条直线是否平行。,AB Parallel with CD?,求出侧面投影,Obtain the profile projection,如何判断？,So how
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