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.

190

DESCRIPTIVE GEOMETRY

the sphere is already determined. Take the plane which is tangent
to the cone along the element B-D as the plane of development.
Then in Fig. 164 the arc F'(-O'l- *. -F,., with center at Bd and
with radius equal to the radius of the sphere, may represent the
indefinite development of the curve cut from the surface of the
cone by the sphere.
To determine the positions in development of the points on this
curve through which the various elements of the cone pass before
development, we must determine the actual distances between
these points measured along the curve before development.
To do this, develop the horizontal projecting cylinder of the
briginal curve, as show in Fig. 163. The base of this cylinder is
f,-o,-q,-. - -f,, and since the plane of the base is perpendicular to
the elements of the cylinder, the curve of the base will roll out
into a straight line to which the elements of the cylinder will
remain perpendicular (see Problem 249, Case 4).
The development of this cylinder is shown in Fig. 165, where
F,-F,/ represents the development of the curve of the base, and
where F-O-Q-.- -F' represents the development of the original
curve of intersection. The actual distances between the points of
division on this curve are now revealed.
Returning now to Fig. 164, start with F', any point on the arc,
and make the distances F'-O'1, OQ-Q, Q'-Gd, etc., such that their
rectified lengths shall be equal respectively to the rectified distances F-O, O-Q, Q-G, etc., of Fig. 165. Through Bd and the
points just determined draw B"l-Fd-Dd, B'-Od-Nd, etc., to represent in development the elements of the cone passing through the
points F, 0, Q, etc., of Fig. 163.
Let it now be required to develop the curve of the base of the
cone. In Fig. 164 make Bd-Dd, Bd-Nd, Bd-Pd, etc., equal respectively to B-D, B-N, B-P, etc., of the original cone, Fig. 163. The
curve D-NX-P'-R`-D' represents the curve of the base of the
cone in development. In the same way we may find the development of any curve on the surface of the cone.
422. Problem 291. Find the intersection of a sphere by a cone
when the axis of the cone contains the center of the sphere, and when
the vertex of the cone is outside the surface of the sphere.