Elements of descriptive geometry, with applications to isometric projection and othe forms of one-plane projection; a text-book for colleges and ingineering schools by O. E. Randall.

RELATING TO POINT, LINE, AND PLANE

57

parallel to V. The plane U cuts the plane S in a straight line
B-E parallel to S-s' and vertically projected in b'-e' parallel to
S-s' (see Sections 30 and 41). The plane U cuts the plane T in the
line D-E vertically projected in d'-e'. The point el in which the
two vertical projections intersect must be the vertical projection of
the point in which the lines B-E and D-E intersect, and is, according to analysis, the vertical projection of a point in the required
intersection. Since the point E is in the plane U, its horizontal
projection e, must be upon U-u,, and therefore at the intersection
of U-u, and a straight line through e' perpendicular to G-L.
Pass another auxiliary plane W parallel to V, and in the same
manner obtain K, another point in the required intersection.
The line A-E-K, whose horizontal projection is a,-e,-k, and
whose vertical projection is a'-e'-k, is the required line.
Check. The straight line E-K thus determined by the two points
E and K should pass through A.
147. Problem 85. Find the intersection of the two planes S =- 2,
330~, 300~, and T= 0~, - 3, x.
148. Problem 86. Find the intersec- t
tion of the two planes S=-2, 60~,
30~, and T= 0~, ac, 2.
149. Problem 87. Find the intersec- d
tion of the two planes S =- 6, 30~, 90~, 9/,T
and T= 6, 135p, 90~. L
150. Problem 88. Find the intersec- cd
tion of the two planes S = 0~, 6, 2, and
T= 0, 2, 6. \
151. Problem 89. To find the point b'
in which a given straight line intersects FIG. 67 t
a given plane.
Analysis. The given line must intersect the given plane in the
line in which any auxiliary plane containing the given line intersects the given plane. The point in which the given line crosses
this line of intersection must be the required point.
CASE 1. When the given plane is located by its traces.
Construction. See Fig. 67. Let M-N represent the given line
and let S represent the given plane.