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

165

CASE 4. To develop the surface.
Analysis and Construction. Take the plane W, Fig. 151, as the
plane of development. Starting with the element of tangency
N-0 in this plane, roll the cylinder along the plane away from V,
bringing the successive elements into contact with W.
Since the plane of the base of the cylinder is perpendicular to
the elements, the curve of the base will roll out into a straight
line perpendicular to the elements.
The character of the development is shown in Fig. 152, where
N-O represents the element of tangency, and where N-N' drawn
perpendicular to N-O and made equal to the rectification of the
curve of the circular base of the cylinder represents the development of the base.
The elements through A, L, D, etc., in Fig. 151, will take the
positions A-K, L-Mf, D-E, etc., in Fig. 152, where N-A, A-L,
L-D, etc., are made equal respectively to the rectified arcs n-a,,
a,-l,, l-d,, etc., of Fig. 151.
The points O, K, M, E, O, Fig. 151, which are situated upon
the elements just named, will in development take the positions
0, K, M, E, Ot in Fig. 152, where N-O, A-K, L-3I, etc., are
made equal respectively to the distances of the same name in
Fig. 151.
The development of the curve of intersection is represented by
O-K-M-E-O'.
The tangent O-Q of Fig. 151 will in development take the
position O-Q in Fig. 152, where N-Q is made equal to n,-q, of
Fig. 151.
378. Problem 250. Given a right circular cylinder whose axis is
perpendicular to H, and given a cutting plane; required to find the
intersection by passing the auxiliary planes parallel to H.
379. Problem 251. Given a right circular cylinder whose axis is
perpendicular to H, and given a cutting plane which is parallel to
G-L but oblique to H and V; required to find the intersection, 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.
380. Problem 252. Solve the above problem when the cutting plane
is perpendicular to V but oblique to H.