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.

150

DESCRIPTIVE GEOMETRY

Produce M-N to intersect H in a, and to intersect V in b', points
in the required traces.
Following Analysis 2, take the point A - the point in which M1-N
pierces I- as the vertex of a cone whose elements are tangent to
the sphere. The lines a,-d, and a,-e,, each tangent to the horizontal projection of the sphere, will represent the horizontal
projections of the extreme elements of the cone. The line a,-c,
represents the horizontal projection of the axis of the cone, and
U-d,-e,-u, represents the horizontal trace of the plane of the circle
of tangency, which circle may be taken as the base of the cone.
Determine the point F in which M-N intersects U. Revolve U
about its horizontal trace U-u, as an axis into H. F will fall at
f,, and the circle of tangency will fall at d,-kH-e,. Through f,
draw fH-kH-l, tangent to the circle d,-ki-e, at the point k,.
This line is, by Section 321, the revolved position of a line of
the plane containing the line M-N and tangent to the cone, and
is therefore a line of the required plane.
In true position this line F-K-L pierces H at 1,. Through a,
and 1, draw the required horizontal trace S-s,. Through S and b'
draw the required vertical trace S-s'.
Check. The required plane is tangent to the cone along the
element A-K, and since this element is tangent to the sphere at
the point A-, K is the point at which the required plane is tangent
to the sphere. The required plane must then be perpendicular to
the radius K-C. The projections of K in true position are k, and
k', where the distance of k' from G-L is equal to k,-kH. Therefore
S-s, and S-s' should be perpendicular respectively to k,-c, and k'-c'.
Since from the point fH another tangent to the circle d,-k11-e,
may be drawn, another plane answering the conditions of the
problem may be constructed.
Construction 3. See Fig. 141. C represents the center of the
sphere and M31-N represents the given line.
M-N pierces H at a, and pierces V at b', points in the required
traces.
Following Analysis 3, take for the vertices of the cones the
points D and E, which are assumed upon M1-N in such a way that
D is at the same distance below H as the center of the sphere,