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

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196. Problem 134. Given the horizontal trace [S=- 4, 3, 0;
s, = 4, 6, 0] of a plane S, and given the angle 60 degrees which the
plane S makes with H; required the vertical trace of the plane S.
197. Problem 135. Given the vertical trace [S=-5, 0, 0;
s'= 3, 0, 5] of the plane S, and given the angle 30 degrees which
the plane S makes with V; required the horizontal trace of the plane S.
198. Problem 136. Given the horizontal trace [S =- 5, - 4, 0;
st = 5,- 4, 0] of the plane S, and given the angle 60 degrees
which the plane S makes with H; required the vertical trace of
the plane S.
199. Problem 137. Given two straight lines not in the same plane;
required to draw a third straight line which shall be perpendicular
to both.
Analysis 1. Through one of the lines pass a plane parallel to
the other line, and project the second line upon this plane. This
projection must be parallel to the second line and intersect the
first line. At this point of intersection erect a straight line perpendicular to the plane.
This line must be perpendicular both to the first line and to
the projection of the second line. It will remain in the projecting plane of the second line and therefore intersect the second
line. It will be perpendicular to the second line since it is perpendicular to the projection of the second line on a plane to which
the second line is parallel. It is therefore perpendicular both to
the first line and to the second line.
Analysis 2. In case we are required to find simply the shortest
distance between two straight lines not in the same plane, we may
proceed as follows: Through any point of one of the lines pass a
plane perpendicular to the other line. Project the first line upon
this plane. The distance from the point in which the second line
intersects the plane to the projection of the first line upon the
plane is the required distance.
Construction. See Fig. 79. Let M-N and O-P represent the
given lines.
Following Analysis 1, draw through M-N the plane S parallel
to O-P. To do this, draw through any point of M-N, as D, the
line D-E parallel to O-P. Find the point e, in which D-E pierces