In Fig. 495, let F L M K represent the section of the cone the pattern for which is required. Produce the sides F L and K M until they meet in the point N, which is the apex of the cone of which F L M K is a frustum. Through N draw N E. bisecting the angle L N M and constituting the axis of the cone, which produce in the direction of D indefinitely. From K draw K H at right angles to the axis. At any convenient distance above the cone construct a plan or profile as it would appear when cut on the line K H, letting the center of the profile fall upon the axis produced, all as shown by A D C B. Divide the profile into any number of equal parts, and from the points thus obtained draw lines parallel to the axis, cutting K H. From the apex X, through the points in K H, draw lines cutting the top L M and the base F K. Place the blade of the T-square at right angles to the axis of the cone, and, bringing it successively against the points in L M and F K, cut the side N F, as shown above L, and from H to F. From N as center, with radius N H, strike the arc T S indefinitely, upon which lay off a stretchout from the plan, as shown, and through the points of which, from the center N, draw lines indefinitely, as shown. With the point of the compasses still at N, and the pencil brought successively against the points in the side from H to F, describe arcs, which produce until they cut corresponding lines drawn through the stretchout. Then a line traced through these points of intersection, as shown by T U S, will form the lower line of pattern. In like manner draw arcs by radii corresponding to the points in the side at L, which produce also until they intersect corresponding lines drawn through the stretchout. A line traced through these points, as R P O, will be the upper line of the pattern sought.
Fig. 495. - The Envelope of a Frustum of a Right Cone Contained between Planes Oblique to its Axis.