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 LINES

157

Take as the auxiliary plane the horizontal projecting plane of
the given line. This plane U cuts the sphere in a small circle
whose center is A and whose diameter is B-D.
To determine the points in which M-N intersects this circle,
revolve the plane U about its horizontal trace into H. The center
A of the circle cut from the sphere falls at aH, and the line 1-N
falls at mH-nH. The points xX and y1, the points in which mH-nH
intersects the circumference of the circle a, are the required points
in revolved position. The points X and Y, whose projections are
(x,, x') and (y,, y') respectively, are the required points in true
position.
The problem may be as easily solved by using the vertical projecting plane of 2M-N as the auxiliary plane.
366. Problem 239. Given a hollow hemisphere; required to find
the points in which a given straight line intersects the inner and the
outer surface.
Analysis. Take as the auxiliary plane one of the projecting
planes of the given line, as in the preceding problem. Find the
two semicircles in which this plane intersects the outer and inner
surfaces of the hemisphere. The points in which the given line
intersects these semicircles will be the required points.