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.

SURFACES OF DOUBLE CURVATURE

149

U will fall at cH. Through dH draw eJ-dH-f, tangent to the circle
cH at el. The straight line of which e1-dd-ff is the revolved position passes through D and is tangent to the great circle cut from
the sphere by U, and is therefore a line of the required plane.
Produce this line to meet H in f,. Through f, and a, draw S-s,,
the required horizontal trace. Through S and b' draw S-s', the
required vertical trace.
Check. Assume any point, as G, in the vertical trace of U. After
the revolution of U, G will take the position g,, and the vertical
trace U-u' will take the position U-g,. The point kg, the point in
which U-gH intersects e,-f,, is the revolved position of the point
in which E-F crosses the vertical trace of U, or, in other words,
I / \
FI.. 140v
FIG. 140
k1, is the revolved position of the point in which E-F pierces V, a
point necessarily in the vertical trace of the required plane.
The true position of this point K must be upon U-u' at a distance from U equal to U-kH, and should be on S-s' already located.
Since from dN another rectilinear tangent to the circle c, may be
drawn, another plane answering the conditions of the problem may
be constructed.
Construction 2. In Fig. 140 let C represent a sphere whose center is in G-L and let M-N represent the given line.