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.

162

DESCRIPTIVE GEOMETRY

CASE 3. To develop the surface.
Analysis. Take the plane of the face A-B-D, Fig. 149, as the
plane of development. Starting with this face in the plane, roll the
pyramid from left to right, bringing first the face A-D-E, second
the face A-E-F, and finally the face A-B-F, all into coincidence
with the original plane.
Construction. Fig. 150 shows how these faces will appear after
development.
The triangle A-B-D, Fig. 150, in which A-B, B-D, and A-D
are equal respectively to A-B,* b,-d,, and A-D of Fig. 149, represents the face A-B-D of Fig. 149. Then since tle face A-D-E,
Fig. 149, is revolved about the edge A-D as an axis, it will
in development take the position A-D-E, Fig. 150, where A-D,
D-E, and A-E are equal respectively to A-D, d,-e,, and A-E
of Fig. 149.
By a similar process the remainder of the figure may be constructed.
The vertices of the polygon of intersection will in development
take the positions X, Y, Z, W, and X' in Fig. 150, where A-X,
A-Y, A-Z, A-W, and A-X' are made equal respectively to A-Xt
A-Y, A-Z, A-W, and A-X of Fig. 149.
The surface A-B-D-E-F-B', Fig. 150. represents the development of the surface of the pyramid. The broken line X- Y-Z- W-X'
represents the development of the line of intersection. The surface X-B-D-E-F-B'-X'- TV-Z- Y represents the development, of
that portion of the surface of the pyramid below the plane S.
In cases of oblique and irregular pyramids the edges of the
pyramid will not be equal, neither will the sides of the base necessarily be equal, as in the case just considered.
374. Problem 246. Given an oblique pentagonal pyramid, and
given a cutting plane perpendicular to V but oblique to H; required
to find the intersection, the true size of this intersection, and the
development of the surface of the pyramid.
* In Fig. 149 the distances A-B, A-D, A-E, etc., are not equal to a'-b', a'-d',
a'-e', etc.
t In Fig. 149 the distances A-X, A-Y, A-Z, etc., are not equal to a'-x', a'-y',
a'-z', etc.