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.

64

DESCRIPTIVE GEOMETRY

Construction. See Fig. 74. Let M-N represent the line through
which the plane is to be passed, and let O-P represent the line to
which the plane is to be parallel.
Through any point of M-N, as i1, draw M-Q parallel to O-P
and produce it to intersect H in c, and to intersect V in d'. Find
the point a, in which Ml-N
intersects H; also find the
i | point b' in which M-N in-,c/n, By^ tersects T. Through t and.L/7', m, —r. a- io a draw S-st, and through b'..\.'. n^-2 " -L and d' draw S-s'. These are
Xm\,\ the traces of the required
plane and should cross
\i, \p,' G-L at the same point.
'.,,d' Check. Through some
other point of MV-N draw
another straight line paralFIG. 74 lel to O-P and note
whether this line intersects H and V in the traces now located.
176. Problem 114. Pass a plane through [M = - 6, 2, 6; N = 4,
- 5, 1] and parallel to [O =1, - 1, 6; P = 1, 5, 2].
177. Problem 115. Pass a plane through [3M=5, - 6, -3;
N=-4, 1, 8] and parallel to [0 =-5, 3, 5; P = 2, 3, 5].
178. Problem 116. Pass a plane through [= — 6, 4, - 6; N
= 5, 4, - 6] and parallel to [0 = 2, 2, 6; P = 2, - 6, 1].
179. Problem 117. To find the shortest distance from a given
point to a given straight line.
The required distance must be measured upon a straight line
drawn from the given point perpendicular to the given line.
Analysis 1. If we pass a plane through the given point and the
given line, and revolve this plane about one of its traces into the
corresponding plane of projection, the line and the point will be
shown in their true relation. From the revolved position of the
point draw a straight line perpendicular to the revolved position
of the line, and upon this line measure the required distance.
Analysis 2. Draw through the given point a plane perpendicular to the given line and find the point in which this plane is