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.

INTERSECTION OF SURFACES BY SURFACES 183

the base of the first cone. This auxiliary plane cuts from the
first cone the two elements B-K and B-L.
The same auxiliary plane intersects the plane S in the line F-G,
which must cross the base of the second cone in points of the
elements cut from the second cone by this auxiliary plane.
To find these elements, revolve S about S-s' into V. The center
of the circular base of the second cone will fall at c", the point F
will fall atfv, and the point G will fall at gv'
The line fv-gv-mV represents the revolved position of F-G, and
the points n" and mV represent the revolved positions of the two
points in which F-G crosses the base of the second cone.
After the counter revolution M and NR will take the positions
(mn,, m) and (n,, n') respectively, where m, and n, are at the same
distances from G-L as M and N respectively are from the axis of
revolution S-s'.
The two elements cut from the second cone by this auxiliary
plane are then D-M and D-N. These elements cross the two elements cut from the first cone by this same plane at O, P, Q, and
R, four points of the required curve of intersection.
Other auxiliary planes may be drawn in the same way, and a
sufficient number of points may be determined to locate the
required curve of intersection.
That which has been said in previous sections regarding the
determination of visible and invisible portions of the curve of intersection, the construction of rectilinear tangents to this curve, and
the development of the surfaces, may be said of these surfaces also.
408. To determine in advance the Nature of the Curves in which
Cylinders and Cones intersect. The nature of the curve of intersection will depend upon the relative size and position of the
intersecting surfaces. The surfaces may intersect so that a portion of one will remain entirely outside of the other, giving
one continuous curve of intersection, as in Fig. 159; or they
may intersect so as to have one distinct curve of ingress and
another distinct curve of egress, as in Fig. 160; or the two surfaces may lie between two planes to which the surfaces are both
tangent, in which case the curve of ingress will be tangent to the
curve of egress on opposite sides of the surfaces, as in Fig. 161.