This subject embraces a large variety of forms of frequent occurrence in sheet metal work, and the development of their surfaces comes under an altogether different set of rules than those applied to parallel forms.

Before entering into the details of these methods it will be best to first define accurately what is here included by the use of the term. These forms include only such solid figures as have for a base the circle or any of the regular polygons, as the square, triangle, hexagon, etc.; also figures though of unequal sides that can be inscribed within a circle, and all of which terminate in an apex located directly over the center of the base.

Fig. 243. - A Right Cone Generated by the Revolution of a Right-Angled Triangle about its Perpendicular.

While the treatment of these forms has been said to be altogether different from that of parallel forms there are some points of similarity to which the student's attention is called that may serve to fix the methods of work in his memory.

Whereas in parallel forms the distances of the various points in a miter are measured from a straight line drawn through the mold near the miter for that purpose, as C D, Fig. 288, the distances of all points in the surfaces of tapering solids produced by the intersection of some other surface are measured from the apex upon lines radiating therefrom; and whereas the distance across parallel forms (the stretchout) is measured upon the profile, the distance across tapering forms is measured upon the perimeter of the base.

Fig. 244 - A Right Cone with Thread Fastened at the Apex to which are Attached Points Marking the Upper and Lower Bases of a Frustum.

Patterns are more frequently required for portions of frustums of these figures than for the complete figures themselves and the methods of obtaining the pattern of coverings of said frustums is simply to develop the surface of the entire cone or pyramid and by a system of measurements take out such parts as are required.

As the apex of a cone is situated in a perpendicular line erected upon the center of its base, it must of necessity be equidistant from all points in the circumference of the base.

In works upon solid geometry the cone is described as a solid generated by the revolution of a right-angle triangle about its vertical side as an axis. This operation is illustrated in Fig. 243, in which it will be seen that the base E D of the triangle C E D is the radius which generates the circle forming the base of the cone, and that the hypothenuse CD in like manner generates its covering or envelope.

If a plane be passed through a cone parallel to the base and at some distance above it, the line which it produces by cutting the surface of the cone must also be a circle, because it, like the base, is perpendicular to the axis. The portion cut away is simply another perfect cone of less dimensions than the first, while the portion remaining is called a frustum of a cone. AFC, Fig. 244, is a cone, and BDEC, Fig. 245, is a frustum. The line B E, Fig. 244, shows where the cone is cut to produce the frustum.

Fig. 245. - Frustum of a Right Cone, the Dotted Lines Showing the Portion of the Cone Removed to Produce the Frustum.

If, having a solid cone of any convenient material, as wood, a pin be fastened at the apex C of the same, as shown in Fig. 244, and a piece of thread be tied thereto, to which are fastened points B and A, corresponding in distance from the apex to the upper and lower bases of the frustum, and the thread, being drawn straight, be passed around the cone close to its surface, the points upon the thread will follow the lines of the liases of the frustum throughout its course. If then, taking the thread and pin from the cone, and fastening the pin as a center upon a sheet of paper, as shown in Fig. 246, the thread be carried around the pin, keeping it stretched all the time, the track of the points fastened to the thread will describe upon the paper the shape of the envelope of the frustum, as shown by G D E F. By omitting the line produced by the upper of the two points, the envelope of the complete cone G C F will be described. The length of the arc G F described by the point A attached to the thread may be determined by measuring the circumference of the base of the cone by any means most available. The usual method is to take between the points of the dividers a small space and step around the circumference of the circle of the base and set off upon the circle of the pattern the same number of spaces.

Fig. 246. - Envelope of the Cone and Frustum Described by the Pin and Thread in Fig. 244.

Fig. 247. - Unfolding the Envelope of a Right Cone.

The development of the envelope of a cone may be further illustrated by supposing that, in the case of the wooden model, it be laid upon its side, upon a sheet of paper and rolled along until it has made one complete revolution; a point having been previously marked upon the line of its base by which to determine the same. The base B, Fig. 247, thus becomes stretched out as it were, describing the line C D upon the paper, while the apex A, having no circumfer-ference, remains stationary at the point A1. The lines C A1 and D A1 represent the contact of the side of the cone at the beginning and at the finish of one revolu tion.