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.

CHAPTER XVII

INTERSECTION OF SURFACES BY SURFACES
404. General Instructions. Pass a series of auxiliary planes so as
to cut lines from the two surfaces. The points in which the lines
of one surface intersect the lines of the other are necessarily in both
surfaces and therefore in their line of intersection.
Pass the auxiliary planes in such a way as not only to cut from
the surfaces the simplest lines'but also in such a way as to make
the work of construction as simple as possible.
If the two surfaces are of revolution, with intersecting axes, it
will often be found convenient to pass a series of auxiliary cutting
spheres with centers at the intersection of the axes. A sphere of
this character will cut from the two surfaces circumferences of
circles which, lying on the surface of the sphere, will as a rule intersect. These points of intersection must lie on the line in which the
given surfaces intersect, since they are common to both surfaces.
405. Problem 275. To find the intersection of a cylinder and a
cone, to draw a rectilinear tangent to the curve of intersection, and
to develop the surfaces.
CASE 1. To find the intersection.
Analysis. Pass the auxiliary planes through the vertex of the
cone and parallel to the axis of the cylinder.
Construction. Let the cylinder be represented with its base on
H, and let the cone be represented with its base on V, as shown in
Fig. 159. The axis of the cylinder is represented by A-B and
the axis of the cone is represented by C-D.
A straight line through the vertex of the cone parallel to the
axis of the cylinder will be common to all the auxiliary planes.
Through D draw D-E parallel to A-B and produce it to meet
H at e, and to meet V at f'. The horizontal traces of all the
auxiliary planes must pass through e,, and the vertical traces of
all the auxiliary planes must pass through f'.
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