SYSTEM OF PROJECTION 



175 



The inclination of the original of this line can be determined; for its 

 depth below the surface at the intersection of the two contours is known, 

 and this depth divided by ca" gives the tangent of the inclination ; that is, 

 tan 8" = a" b" /ca" = ab/ca" ; or we may determine the inclination of 

 the line graphically by drawing a" b" = ab at right angles to cd, and the 

 inclination 8 is found by completing the triangle ca" b" . 



To determine the point where a straight line, whose projection is V c' , 

 figure 3, pierces a given plane, t, pass a plane through the line and let. 

 its trace be parallel with that of the given plane. This trace will pass 

 through the point where the line cuts the reference plane. Draw a 

 straight line from this point at right angles to the two traces and it will 

 make an angle, 6, with the projection of the given line ; let 8' be the dip 

 of the line, and 8" that of the plane we have passed through it; then 

 tan 8" — tan 8' cos 0; or we may de- 

 termine 8" graphically as follows: 

 Lay off the inclination 8' of the line, 

 and from any point, d', of its pro- 

 jection, draw d' e', which will be its 

 depth at that point. Draw d' d" 

 parallel with the trace, t', and it will 

 be the projection of a contour on 

 the plane f ; the depth of the plane 

 t' at d" will be d" e" = d' e' ; draw 

 d" e" and complete the triangle t' d" 

 e", and the dip of the plane, t', will 

 be 8". Lay off the dips, 8 and 8" at 

 t and t', and produce the lines until 

 they intersect in c. The two planes 

 will meet on the contour through c, 

 whose depth, be = tb tan 8 = bV tan 8". The given line lying in the plane 

 t' will intersect the same contour under c f , which is therefore the projec- 

 tion of the point where it intersects the plane t. If the plane is the 

 reference plane and the line is given by its horizontal projection and the 

 depths of two points on it, such as the line ff', figure 7, we lay off these 

 depths, fg w and /' g m , at right angles to the line, and draw g w g m ; it in- 

 tersects the line ff' in the horizontal reference plane at h; for if we rotate 

 the triangle, hfg iv , about ff' until it lies in a vertical plane, g w g m will 

 be the true position of the original line ; or we may determine k by means 

 of the proportion lef :ff = f g m = fg iv — f g m . 



To find the point where a horizontal line pierces a plane, draw the 

 projection of the contour on the plane at the same depth as the line ; the 



Figure 3.- 



-1 nter section of a Line and 

 a Plane 



