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.

184

DESCRIPTIVE GEOMETRY

The nature of the curve of intersection may be determined in
advance by drawing, under the same conditions as the auxiliary
planes are drawn, tangent planes to the surfaces.
If in Fig. 159 we draw, as one of the auxiliary planes of this
problem, the plane T tangent to the cylinder along the upper surface, the position of its vertical trace T-t' shows that such a plane
intersects the cone, and that a portion of the cone remains entirely
outside the cylinder.
The plane TV drawn under the conditions mentioned above and
tangent to the cylinder along the under surface does not cut the
cone, showing that a portion of the cylinder remains outside the
cone.
The curve of intersection then in this case will be one continuous curve, as shown in the figure.
If both the planes T and WT had intersected the cone, the indication would be that the cylinder passed through the cone, giving
two distinct curves of intersection.
If the base of the cone had fallen wholly within the two vertical
traces T-t' and W-w', the indication would be that the cone passed
through the cylinder, giving two distinct curves of intersection.
This last condition is illustrated in Fig. 160, where the cylinder
whose axis is C-D intersects the cylinder whose axis is A-B in
two distinct curves. The base of the cylinder whose axis is C-D
falls wholly within the two planes T and U, which are drawn
under the same conditions as the auxiliary planes of this problem,
and tangent to the cylinder whose axis is A-B.
If the two vertical traces T-t' and TW-w' in Fig. 159 had included
and had been tangent to the base of the cone, the indication would
be that the two surfaces were included by the tangent planes bringing the curves of intersection into a position of tangency. This
condition is illustrated in Fig. 161, where the first cone whose axis
is A-B intersects the second cone in two tangent curves. In
Fig. 161 the two planes whose intersections with the plane of the
base of the first cone are E-U and E-TW are tangent to the first
cone and are also tangent to the second cone, as may be seen from
the revolved position of the lines in which these tangent planes
intersect S, the plane of the base of the second cone.