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

37

Through o,, the horizontal projection of 0, draw o,-a, perpendicular to m-n,. This line must be the horizontal trace of the
plane containing 0 and perpendicular to the axis, and therefore
the plane in which 0 moves during revolution.
After revolution the point 0 must fall somewhere upon a,-o,
produced, and at a distance from the axis equal to its original
distance from the axis.
This distance is O-a,, which is the hypotenuse of the rightangled triangle O-o,-ar. The base of this triangle is equal to the
distance of the horizontal projection of the point 0 from the axis,
and the altitude is equal to the distance of the point 0 from H,
or, what is the same thing, the distance of the vertical projection
of the point 0 from G-L. Therefore lay off from a,, upon a,-o,
-- o. --- —--
1 /
FIG. 44 FIG. 45
produced, the distance a,-o. equal to a,-0. The point oI will be
the revolved position of 0.
Construction. Fig. 45 shows how the work is done when the
planes H and V occupy their position of coincidence. The line
o,-a, is drawn through o, perpendicular to m,-n,, and a,-o,- is made
equal to the hypotenuse of a right-angled triangle whose base is
equal to o,-a, and whose altitude is equal to o'-b.
This hypotenuse is calculated graphically in the triangle c-d-e,
where c-d equals o,-a, and where c-e equals b-o'.
It will be noticed in Fig. 44 that if the point 0 is so situated
that its horizontal projection falls on m,-n,, the base of the triangle
will vanish and the hypotenuse will be equal to the altitude. In
such a case then the distance of the point from the axis, or the