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.

156

DESCRIPTIVE GEOMETRY

361. Problem 234. To find the points in which a given straight
line intersects the surface of a cone.
Analysis. Pass the auxiliary plane through the line and the
vertex of the cone.
XYt~ ^Construction. Let the cone be
\a \ represented as in Fig. 145, and let
/f_ ---.~T,^ /\ M-N represent the given line.
i\ \ ' n\ Produce M-N to pierce H at b,.
Through A, the vertex of the cone,
G,;J, _ eXL \ / \and parallel to the given line, draw
A-D piercing H at d,. U-d,-b,-u,
i s the horizontal trace of the aux'/\ /i \ iliary plane. This plane intersects
/m aN'\ the surface of the cone in two elements, A-E and A-F, which are
U cut by M-N in X and Y, the two
FIG. 145 points required.
362. Problem 235. Find the points in which a given straight
line intersects the surface of an inverted cone situated in the third
quadrant.
363. Problem 236. Find the points in which a given straight line
intersects the surface of the frustum
of a cone.
364. Problem 237. Given a hollow e,
cone; required to find the points in /
which a given straight line intersects
the inner and the outer surface of 4
the cone.,yn\
365. Problem 238. To find the G- k \ -- L
points in which a given straight line
intersects the surface of a sphere.
Analysis. Since the intersection
of a sphere by any plane is a circle,
the auxiliary plane may be passed
through the line at random. FIG. 146
Construction. Let the sphere be represented as in Fig. 146, and
let M-N represent the given line.