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.

IN'TERSECTION OF SURFACES BY SURFACES 187

CASE 2. To draw a rectilinear tangent to the curve of intersection.
The rectilinear tangent to the curve at any given point will be
the intersection of two planes, one tangent to the cylinder at the
given point and the other tangent to the sphere at the same point.
417. Problem 286. Find the intersection of a sphere and a cylinder
when the axis of the cylinder is vertical and passes through the center
of the sphere.
418. Problem 287. Find the intersection of a sphere and a cylinder
when the axis of the cylinder is horizontal and passes through the
center of the sphere.
419. Problem 288. Find the intersection of a sphere and a cylinder
when the axis of the cylinder does not pass through the center of the
sphere, and when some of the elements of the cylinder remain wholly
outside the sphere.
420. Problem 289. To find the intersection of a hemisphere and a
cone and to draw a rectilinear tangent to the curve of intersection.
CASE 1. To find the intersection.
Analysis 1. Pass the auxiliary planes through the vertex of the
cone and perpendicular to H, cutting elements from the cone and
semicircles from the hemisphere.
Analysis 2. Same as Analysis 2 of Section 416.
Construction. Let the cone and the hemisphere be represented
as in Fig. 163, where the vertex of the cone is taken at the center
of the hemisphere.
The plane S represents one of the auxiliary planes, cutting the
cone in two elements B-R and B-P, and cutting the hemisphere
in a semicircle.
To find the points in which these elements intersect the semicircle, revolve S about a vertical axis through B until it is parallel
to V. The two elements B-R and B-P will then be vertically
projected at b'-r" and b-p" respectively, and the vertical projection
of the semicircle will be coincident with the vertical projection of
the hemisphere. Therefore u" and q" are the vertical projections
of the revolved positions of two points of the required curve of
intersection.
The auxiliary plane T, parallel to V, cuts the cone in two
elements B-K and B-L, and cuts the hemisphere in a semicircle