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 SURFACES 181

A plane through one of the axes and parallel to the other will
itself be one of the auxiliary planes, and will be parallel to all the
other auxiliary planes.
Through any point on the axis C-D, as D, draw D-E parallel
to the other axis A-B. C-D pierces H at c, and D-E pierces H
at e,. S-c,-e,-s, is the horizontal trace of an auxiliary plane, and
one to which all the other auxiliary planes must be parallel.
The plane S cuts the cylinder whose axis is A-B in two elements, F-G and K-L, and cuts the other cylinder in two elements,
M-N and 0-G. These elements cross in the points N, G, L, and
P, four points in the required curve of intersection.
Other auxiliary planes parallel to S may be drawn, and a sufficient number of points may be located to determine the curve of
intersection.
If it is desired to determine a point upon any particular element
of either cylinder, we have but to pass the auxiliary plane so as
to cut from the cylinder this element.
When projecting on H, that portion of the curve of intersection
which lies on the upper surfaces of both cylinders and between
extreme elements will be visible.
When projecting on V, that portion of the curve of intersection
which lies on the front surfaces of both cylinders and between
extreme elements will be visible.
CASE 2. To draw a rectilinear tangent to the curve of intersection
at a given point.
Analysis. See Problem 275, Case 2. The required tangent will
be the intersection of two planes, - one tangent to one cylinder
at the point in question, and the other tangent to the other cylinder
at the same point. Therefore draw two planes under these conditions and find their line of intersection.
CASE 3. To develop the surfaces.
The surfaces of the cylinders and the curves of intersection
which are common to both surfaces may be developed by rules
previously given (see Problem 253, Case 4).
407. Problem 277. To find the intersection of two cones, to draw
a rectilinear tangent to the curve of intersection, and to develop the
surfaces.