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.

120

DESCRIPTIVE GEOMETRY

If through E we draw another straight line G-E-K parallel to V
and making the same angle with H as D-E-lF and revolve this
line about the axis A-B, it will generate a surface identical with
the one which we. have been considering, since for any given distance above or below the plane of the circle of the gorge points
upon the two generatrices are equidistant from the axis.
The surface can then be generated by the revolution of two distinct straight lines. Therefore through any point of the surface
two rectilinear elements of the surface may be drawn.
Since every element of the surface intersects the circumference
of the circle of the gorge, and since the circle of the gorge is the
smallest circle of the surface, the horizontal projection of each
element of the surface must contain a point in the horizontal
projection of the circumference of the circle of the gorge and yet
cannot intersect it. Therefore the horizontal projection of each
element of the surface must be tangent to the horizontal projection of the circle of the gorge.
298. To assume a Rectilinear Element of the Hyperboloid of Revolution of One Nappe. See Fig. 118. Assume any point, as P, upon
the base. Through p, draw p,-o, tangent to the horizontal projection of the circle of the gorge. This is the horizontal projection
of an element of the surface piercing H at P and crossing the
circle of the gorge at O. P is vertically projected at p' and 0 is
vertically projected at o'. Therefore p'-o' is the vertical projection
of the element in question.
299. To assume a Point upon the Surface of the Hyperboloid
of Revolution of One Nappe. First assume an element of the
surface and then assume a point upon the element.
Or we may assume the horizontal projection of the point at
random, through it draw the horizontal projection of the element
containing the point, thence the vertical projection of the same
element as explained above, and finally the vertical projection of
the point upon the vertical projection of the element.
300. To determine the Meridian Curve of the Hyperboloid of
Revolution of One Nappe. We shall consider the section of the
surface made by the meridian plane parallel to V, for the vertical
projection of this curve will be equal in every respect to the curve