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.

THE POINT, LINE, AND PLANE

9

other, and so that the two projections of each point shall fall upon
the same straight line perpendicular to G-L, the result will be
precisely the same as that expressed in Fig. 3.
It will be observed that whether we revolve H into coincidence
with V, or V into coincidence with H, or whether the projections
are made upon H and V independently and afterwards combined
as shown above, the result is the same.
The object of the transformation is to make it possible to represent upon a single plane, projections which belong primarily upon
two planes perpendicular to each other.
It is immaterial by which method the transformation is made;
the essential thing is that the student shall be able to pass in imagination, without any difficulty or hesitation, from the position of
perpendicularity to that of coincidence, and vice versa.
Returning now to Fig. 3, which expresses the common result of
the three methods of transformation, it will be noticed that when
a point, as il, is in the first quadrant, its horizontal projection will
be in front of G-L and its vertical projection will be above G-L;
that when a point, as V, is in the second quadrant, its horizontal
projection will be back of G-L and its vertical projection will be
above G-L; that when a point, as 0, is in the third quadrant, its
horizontal projection will be back of G-L and its vertical projection will be below G-L; that when a point, as P, is in the fourth
quadrant, its horizontal projection will be in front of G-L and its
vertical projection will be below G-L.
Conversely, referring to Fig. 3, if a horizontal projection, as m,,
is situated in front of G-L, it will be known that the point M
must be in front of V, that is, either in the first quadrant or in the
fourth quadrant; and if the vertical projection, as mn', of the same
point M is situated above G-L, it will be known that the point 2M
must be above H, that is, either in the first quadrant or in the
second quadrant, and therefore it will be known that the point M
must be in the first quadrant.
If a horizontal projection, as n,, is situated back of G-L, it will
be known that the point N must be back of V, that is, either in
the second quadrant or in the third quadrant; and if the corresponding vertical projection, as n', is situated above G-L, it will be