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.

INTEPSECTION OF SURFACES BY SURFACES 191

423. Problem 292. Find the intersection of a sphere and a cone
when the vertex of the cone lies outside the surface of the sphere, and
when some of the elements of the cone lie wholly outside the sphere.
424. Problem 293. Given a sphere and a point without the surface; required to determine a cone with vertex at the point and
with surface tangent to the sphere, and to determine the circle of
tangency between the two surfaces.
425. Problem 294. To find the intersection of any two surfaces
of revolution when their axes intersect, and to draw a rectilinear
tangent to the curve of
intersection.
CASE 1. To find
the intersection.
Analysis. Draw a
series of auxiliary f| bd, oi Jg, e, —
spheres with centers \
at the point in which \
the axes intersect.
These spheres will
cut from each surface G L
the circumference of [
a circle whose plane d'
will be perpendicular
to the axis of the sur- \
face. The circumfer- '
ences of these circles \i /
will intersectin points \ I I/'
common to both surfaces, and therefore in q ' \
their curve of inter-
section. \ l
Construction. Let it
be required to find the FIG. 166
intersection of an inverted right circular cone by a right circular
cylinder, as represented in Fig. 166. A-B represents the axis of
the cone and D-E represents the axis of the cylinder, the two
axes intersecting at D and lying in a plane parallel to V.