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.

REPRESENTATION OF WARPED SURFACES

107

Under this class of surfaces one, two, or all the directrices may
be rectilinear, or one, two, or all the directrices may be curvilinear.
268. The Hyperboloid of One Nappe. The hyperboloid of one
nappe is a warped surface with three rectilinear directrices.
This surface is represented by locating the three rectilinear
directrices and a sufficient number of the rectilinear elements, or
positions occupied by the generatrix, to reveal the character of
that portion of the surface under consideration.
269. To assume a Rectilinear Element of the Hyperboloid of One
Nappe. See Fig. 112. Let MI-N, O-P, and Q-R represent the three
rectilinear directrices.
Assume any point,
as A, upon the directrix / M,
MJ-N. Assume any two
points, as B and D,
upon the directrix
O-P. Connect A and B, ' kA-E'
and also A and D, by
straight lines and pro- _' J x
duce. The plane of i'
these two lines contains:
the point A and the \ f:l \k'
directrix O-P. By Sec- ' r
tion 151 find the point' \
G in which the remaining directrix Q-R intersects the plane of the two
FIG. 112
lines A-B and A-D.
Connect A and G by a straight line which intersects O-P at K,
since A-G and O-P are in the same plane. A-K-G is an element
of the surface, since it touches the three directrices.
The two points k, and k' should be in the same straight line
perpendicular to G-L, which serves as a check upon the work.
270. To assume a Point upon the Surface of the Hyperboloid of
One Nappe. First assume an element of the surface and then
assume a point upon the element, or follow the directions of
Section 255,