Let K F H G in Fig. 513 be the elevation of a cylinder, which is to be intersected by a. cone or frustum, D A J C, at the angle F D A in elevation, and which is to be set to one side of the center, all as shown by SPLMR of the plan. Opposite the end of the frustum, in both elevation and plan, draw a section of it, as shown by T U V W in the elevation and T1 U1 V1 W1 in the plan. Divide both of these sections into the same number of equal parts, commencing at corresponding points in each, and number them as shown by the small figures in the diagram. From the points in T U V W carry lines parallel to the axis of the cone, cutting the line A J, and thence drop them vertically across the plan. From the points in the section T1 U1 V1 W1 draw lines parallel to the axis of the cone, as seen in plan, intersecting the lines of corresponding number dropped from A J just described. Through these points of intersection trace a line, as shown by L M. Then L M will show the end of the frustum A J as it appears in plan. From X, the apex of the cone in elevation, drop a line vertically, cutting the axis of the cone in plan as shown at X'. From X draw lines through the points in A J and extend them through the side of the cylinder indefinitely. From X1 through the points in L M draw lines cutting the plan of the cylinder, as shown from P to R, and from these points carry lines vertically, intersecting those of corresponding number in the elevation drawn from the apex X. Then a line traced through these points, as shown by D K C N, will be the miter line in elevation.

For the pattern of the frustum, from X as center, with radius X A, describe the arc A1 J', upon which lay off a stretchout of the section T U V W, through the points in which, from X, draw radial lines indefinitely. From the points in D KCN carry lines at right angles to the axis of the cone, cutting the side

A D extended, as shown from d to B. From X as center, with radii corresponding to the distance from X to the various points in the line D B, describe arcs cutting radial lines of corresponding number in the pattern. Through the points of intersection in the pattern thus obtained trace a line, as shown by C1 K1 N1 D1. Then D1 N1 K1 C1 A1 J1 will be the pattern of the frustum D A J C, mitering with the cylinder at the angle described.

Fig. 513.   The Patterns of the Frustum of a Cone Joining a Cylinder of Greater Diameter than Itself at Other than Right Angles the Axis of the Frustum Passing to One Side of that of the Cylinder.

Fig. 513. - The Patterns of the Frustum of a Cone Joining a Cylinder of Greater Diameter than Itself at Other than Right Angles the Axis of the Frustum Passing to One Side of that of the Cylinder.

The method of obtaining the pattern of the cylinder is analogous to that described in the preceding problem, and is clearly shown at the left in the drawing.