Fig. 256. - The Same Plan Shewing Two Sectors of the Smaller Cone in Position Joining the Larger One.
By thus resolving the solid from which the ordinary elliptical flaring article is cut into its component elements the process of developing its pattern may be more readily understood. This process may now be easily explained by returning to the string and pin method which was made use of in connection with the simple cone in the earlier part of this section.
In Fig. 257 is shown a line some distance above the base representing the top of the frustum shown by K L in the original elevation, Fig. 253. It also shows a pin fastened at the apex of the middle conical sector to which is attached a thread carrying points G and H representing the upper and lower surfaces of the frustum. Now, if the string be drawn tight and passed along the side of the larger sector of the cone from A to B the points will follow the upper and lower bases of the frustum. When the point B is reached, if the finger be placed upon the thread at the apex of the lesser cone, shown at C, and the progress of the thread be continued, the points will still follow the lines of the bases of the frustum. If the pin and thread be taken from the cone and transferred to a sheet of paper, as shown in Fig. 259, the pin A being used as a center and the thread as a radius, the points will describe the envelope of the frustum. First, the radius is used full length, as shown by A L K, and arcs L M and K H are drawn in length respectively equal to their representatives H G V and E C W of the original plan. Fig. 253. Then a second pin is put through the string, as shown at B, thus reducing the radius to the length of the side of the lesser cone, and arcs arc struck in continuation of those first described, making the length of the additional arcs equal to those of their corresponding arcs H I J and E A X of the original plan.
Fig. 257. - Opposite View of Parts Shown in Fig. 256, with String Attached to a Pin at the Apex of the Larger Sector.
Fig. 258. - The Completed Solid of which the Ordinary Elliptical Flaring Article is Port.
As the lengths of the sides of the larger and smaller cones above made use of are by construction equal to J P and Z P, the hypothenuses of the triangles, Fig. 254, by whose revolution they were generated, those distances may therefore be taken at once from that diagram by means of the compasses and used as shown in Fig. 259.
Fig. 259. - The Pin and Thread taken from Fig. 257 and Used in, Describing the Envelope.
Reference has been made above to the difference between the circumference of the circle of the base obtained by means of the points and spaces (which method becomes a necessity to the pattern cutter) and the real circumference. An explanation of this difference will lead to the next class of regular tapering figures - viz.: pyramids.
Fig. 260. - An Are Compared with its Chord.
In the accompanying diagram, Fig. 260, ABC represents the arc of a circle of which the straight line A C is the chord, being the shortest distance between the two points A and C. Therefore, when dividing a circle by means of points for purposes of measurement, the pattern cutter is in reality using a number of chords instead of the arcs which they subtend.
In the practice of obtaining the circumference or stretch-out of a circle the space assumed as the unit of measure should be so small that there is no perceptible curve between the points and, of course, no practical difference between the length of the chord and the length of the arc.
It will thus be seen that the circle representing the base of a cone has in reality become in the hands of the pattern cutter a many sided polygon and that the cone is to him a many sided pyramid. As one of the conditions in describing a regular polygon is that its angles shall all lie in the same circle, so the angles or hips of a pyramid must lie in the surface of the cone whose base circumscribes the base of the pyramid and whose apex coincides with the apex of the pyramid. Viewed in this light then, the lines which were drawn upon the outside of the wooden cone for the purpose of measurement in the illustration used above become the angles or hips of a pyramid and may be used for that purpose in exactly the same manner.
In developing the pattern of a frustum of a cone the line connecting the points between G and II, Fig. 251, is supposed, of course, to be a curved line, while in the case of a pyramid (the points or angles of the pyramid being further apart and the sides of a pyramid being flat instead of curved) the lines of the pattern connecting the points would be straight from point to point.