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 185

409. Problem 278. Find the intersection of a cylinder and a cone
when the bases of both surfaces are on V.
410. Problem 279. Find the intersection of two cylinders when the
base of one surface is on H and the base of the other is on V.
411. Problem 280. Find the intersection of two cones when the bases
of both surfaces are on H.
412. Problem 281. Find the intersection of two cylinders of the
same diameter, one horizontal and the other vertical, axes intersecting.
413. Problem 282. Find the intersection of two cylinders of unequal
diameters, one horizontal and the other vertical, axes intersecting.
414. Problem 283. Find the intersection of two cylinders of unequal
diameters, one horizontal and the other vertical, axes not intersecting.
415. Problem 284. Find the intersection of two cylinders of unequal diameters, the larger vertical, the smaller inclined at an angle
of 60 degrees to H, axes intersecting.
416. Problem 285. To find the intersection of a sphere and a
cylinder and to draw a rectilinear tangent to the curve of intersection
at a given point on the curve.
CASE 1. To find the intersection.
Analysis 1. Pass the auxiliary planes parallel to the axis of the
cylinder, cutting from the cylinder elements and cutting from
the sphere circles.
Analysis 2. If the base of the cylinder is a circle, we may pass
the auxiliary planes parallel to this base, cutting circles from both
surfaces.
Construction. In Fig. 162 let C represent the center of the
sphere and let A-B, passing through the center of the sphere,
represent the axis of the cylinder.
Pass the auxiliary planes parallel to A-B and perpendicular
to H. The horizontal trace of the auxiliary plane containing the
axis is S-s,. This plane cuts the sphere in a great circle and cuts
the cylinder in two elements which intersect the circumference of
the circle in four points of the required curve of intersection.
To find these points, revolve the plane S about S-s, into H.
C will fall at cm and the great circle will fall at eH-gH-lH-kH. The
axis A-B will fall at a,-b, and the two elements in which the
auxiliary plane intersects the cylinder will fall at d,-e, and f,-g-,