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.

6

DESCRIPTIVE GEOMETRY

line P-p" is perpendicular to V and is called the vertical projecting
line of P.
It will be noticed in Fig. 1 that the horizontal and vertical projecting lines of a point determine a plane which is perpendicular
to both H and V and is therefore perpendicular to G-L; also that
the straight lines in which this plane intersects H and V are perpendicular to G-L at the same point and pass respectively through
the horizontal and vertical projections of the point.
Observe that when the horizontal and vertical projections of a
point are given, the point itself is definitely located, for the horizontal and vertical projecting lines, determined by the projections of
the point, lie in the same plane and intersect at the only point which
can have its horizontal and vertical projections at the points given.
Observe that the distance of a point from H is in each case
indicated by the distance of its vertical projection from G-L, and
that the distance of the point from V is in each case indicated by
the distance of its horizontal projection from G-L.
If a point is situated in H, its horizontal projection is the point
itself and its vertical projection is in G-L.
If a point is situated in V, its vertical projection is the point
itself and its horizontal projection is in G-L.
If a point is situated in G-L, both its horizontal and vertical
projections coincide with the point itself.
14. Representation of the Point upon H and V in their Position of
Coincidence. In Fig. 2 the projections m,,, "; n,,, n"; o,,, o; and
p,, p" are those previously found in Fig. 1 and represent points in
the first, second, third, and fourth quadrants respectively.
If the plane H be revolved about G-L as an axis until that
portion of H back of V falls on V above H, and that portion of H
in front of V falls on V below H, the point m" will remain stationary, while the point m,, will move in the arc of a circle with a as
a center, and fall at m, in the line m"-a produced.
The point n" will remain stationary, while the point n,, will
move in the arc of a circle with b as a center, and fall at n, in the
line b-n" produced. The point o" will remain stationary, while
the point o, will move in the arc of a circle with d as a center,
and fall at o, in the line o"-d produced.