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.

SURFACES OF SINGLE CURVATURE

133

317. Problem 193. To draw a plane tangent to a cone at a point
on the surface.
Analysis. Since the surface of a cone is of single curvature, the
rectilinear element of the surface through the point of tangency is
the element of tangency, and is therefore a line of the required
plane (see Section 226).
Since the required plane will be tangent to the surface all along
this element, a line tangent to the surface at any point on this
element will be a line of the required plane.
CASE 1. When the base of the cone is a circle on H.
Construction. In Fig. 128 let A-B represent the axis of the cone,
and let 0, assumed as in Section 242, represent the point on the
surface.
The element B-O-D, which is a line of the required plane, pierces
H at d, and pierces V at e', points in the required traces. Through
d,
ia
FIG. 128
d, draw S-d,-s, tangent to the circular base at d,. This is a line of
the required plane, because it is tangent to a curve of the surface
at a point on 0-D. Since this tangent is also in H, it is the required
horizontal trace. Connect S and et by a straight line and produce
it to represent the required vertical trace.
Check. Through F, any point on the element of tangency 0-),
draw F-G parallel to s,-S. This line must be a line of the required
plane, since it is drawn through a point F in this plane and parallel
to a line s,-S also in this plane, and should therefore pierce V in
the vertical trace already found.