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 PLANES

163

375. Problem 247. Given a triangular pyramid, and given a cutting plane parallel to G-L but oblique to Hand V; required to find
the intersection, the true size of this intersection, and the development
of the surface of the pyramid.
376. Problem 248. Given an oblique hexagonal pyramid, and given
a cutting plane parallel to H; required to find the intersection, the
true size of this intersection, and the development of the surface of
the pyramid.
377. Problem 249. To find the intersection of a right circular
cylinder whose axis is perpendicular to H by a plane; to draw a
rectilinear tangent to the curve of intersection; to find the true size
of the intersection, and to develop the surface of the cylinder.
CASE 1. To find the intersection.
Analysis 1. Pass a series of auxiliary planes parallel to the axis
of the cylinder, cutting from the cylinder elements and cutting
from the plane straight lines.
Analysis 2. Pass a series of auxiliary planes perpendicular to
the axis, cutting from the cylinder circles and cutting from the
plane straight lines.
Construction. Let the cylinder whose axis is assumed perpendicular to H be represented as in Fig. 151, and let S represent
the cutting plane.
Pass an auxiliary plane T through the axis and perpendicular
to S-s,. T cuts the cylinder in two elements, A-B and D-E, and
intersects S in a straight line F-G. F-G crosses A-B and D-E at
K and E respectively, two points of the required curve of intersection.
Owing to the position which the plane T occupies with reference
to S-s,, the points K and E must be respectively the highest and
the lowest points of the curve of intersection.
Pass another auxiliary plane U perpendicular to H and parallel
to S-s,. The plane U cuts the cylinder in two elements, L-MJ and
N-0, and intersects the plane S in the straight line P-O-M. The
line P-O-M crosses L-M and N-0 at M and O respectively, two
more points of the required curve.
By passing other auxiliary planes parallel to U we may obtain
a sufficient number of points to locate the curve.