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

161

373. Problem 245. To find the intersection of a square pyramid by
a plane, to find the true size of the intersection, and to develop the
surface of the pyramid.
CASE 1. To find the intersection.
Analysis. Pass the auxiliary planes through the edges of the
pyramid.
Construction. In Fig. 149 let A-B-D-E-F represent the pyramid and let S represent the cutting plane.
Pass an auxiliary plane T through the two edges A-B and A-E.
This plane intersects S in G-K, which crosses the edges A-B and
A-E at X and Z respectively.
'sVt h Pass another auxiliary plane U
\//, \ through the two edges A-D and
A-F. U intersects SinL-O, which,:'K,6 —; \-, \ crosses the two edges A-D and A-F
\^. f \ \ at Yand W respectively. X-Y\d \ Z- W is the required intersection.
b\~-,2::-^.\ \ ~In this particular
G- T("- a, / 9' L case, on account of
t/h 0 il, athe character and
B X A
z,,e' -Vt
~Y
'W\ / B
* FIG. 149 FIG. 150
position of the pyramid, the auxiliary planes T and U are perpendicular to H, but the method of construction is not changed when
the conditions are otherwise.
CASE 2. To find the true size of the intersection.
Revolve the plane S in Fig. 149 about S-s' into V. The points
X, Y, Z, and W will fall at xv yv, zT, and wv respectively (see
Section 91), and xV-yv-zV-wv will represent the true size of the
intersection.