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.

172

DESCRIPTIVE GEOMETRY

of a right circular cone is cut by a plane perpendicular to the axis,
the curve of intersection is the circumference of a circle; and when
cut by a plane containing the axis, the intersections are elements.
By examination of the results obtained in Problems 261-263,
and by reference to treatises on solid and analytic geometry, it
may be shown that when a right circular cone with axis perpendicular to H is cut by a plane making a smaller angle with H than
the elements of the cone, the curve of intersection is an ellipse;
when cut by a plane making the same angle with H as the elements
of the cone, the curve of intersection is a parabola; and when cut
by a plane making a greater angle with H than the elements of
the cone, the curve of intersection is a hyperbola.
393. Problem 264. To find the intersection of any cone by a plane,
to draw a rectilinear tangent to the curve of intersection, and to find
the true size of the intersection.
CASE 1. To find the intersection.
Analysis and Construction. Let the cone be represented as in
Fig. 157, and let S represent the intersecting plane.
Pass the auxiliary planes through the vertex B and perpendicular
to H. Then all the auxiliary planes will intersect in a common
straight line through the vertex and perpendicular to H. This
straight line will intersect S at A (see Section 151), which is therefore a point common to all the lines cut from S by the auxiliary
planes.
Let T represent one of the auxiliary planes. This plane intersects the surface of the cone in two elements, B-D and B-E, and
intersects the plane S in the line F-A, passing through the point
A previously located. F-A crosses B-D and B-E at K and L
respectively, two points in the required curve of intersection.
Another auxiliary plane U will intersect the surface of the cone
in the elements B-M and B-N, and will intersect the plane S in
the line I-A. I-A crosses B-M and B-N at 0 and P respectively,
two more points in the required curve.
CASE 2. To draw a rectilinear tangent to the curve of intersection.
Analysis. See Problem 249, Case 2.
Construction. See Fig. 157. Let it be required to draw a rectilinear tangent to the curve at the point L. Draw the horizontal