In Figs. 526 and 527 are shown perspective representations of scalene or oblique cones. In Fig. 526 the inclination of the axis to the base is so great that a vertical line dropped from its apex would fall outside the base, while in Fig. 527 a perpendicular from its apex would fall at a point between the center and the perimeter of its base.

Supposing the circumference of the base in either case to be divided into a number of equal spaces, it is plain to be seen that lines drawn upon the surface of the cone from the points of division to the apex would be straight lines of unequal lengths, and that such lines would divide the surface of the cone into triangles whose vertices are at the apex of the cone and whose bases would be the divisions upon the base of the cone. It will be seen further that with the means at hand of determining the lengths of these lines forming the sides of the triangles, the pattern cutter possesses all that is necessary in developing their envelopes or patterns.

Problem 163 The Envelope Of A Scalene Cone 533

Fig. 526.

Problem 163 The Envelope Of A Scalene Cone 534

Fig. 527.

Scalene Cones of Different Inclinations.

In Fig. 528, D A H is an elevation of the cone shown in Fig. 526 and D G H is a half plan of the same, drawn for convenience, so that D H is at once the base line of the elevation and the center line of the plan. Fig. .529 shows an elevation and plan of the cone shown in Fig. .527. drawn in the same manner. The principle involved in the development of the patterns of the two oblique cones is exactly the same and. as will be seen, letters referring to similar parts in the two drawings are the same; therefore the following demonstration will apply equally well in either ease. From the apex A drop a perpendicular to the base line, locating the point N. Divide the base D G H into any convenient number of equal spaces, as shown by the small figures, and from the points thus obtained draw lines to the point N. These lines will form the liases of a series of right angled triangles of which A N is the perpendicular hight, and whose hy-pothenuses when drawn will give the correct length of lines extending from the points of division in the base of the cone to the apex. The most convenient method of constructing these right angled triangles is to transfer the distances from N to the various points upon the circumference of the base to the line N D as a base line, measuring each time from the point N, by which method the line A N becomes the common perpendicular of all the triangles. Therefore from N as center, with the distances N 1, N 2, etc., as radii, describe arcs as shown in the engraving, cutting the base line N D. Lines from each of these points to the apex, as A 1, A 2, etc., will be the required hypoth-enuses.

Fig. 528.   Pattern of Cone Shown in Fig. 526.

Fig. 528. - Pattern of Cone Shown in Fig. 526.

The simplest method of developing the pattern is to first describe a number of arcs whose radii are respectively equal to the various hypothenuses just obtained; therefore place one foot of the compasses at A, and, bringing the pencil point successively to the points 1, 2, 8, etc., upon the line N D. describe arcs indefinitely. From any point upon the arc drawn from point 1, as n, draw a line to A as one side of the pattern. Next take between the feet of the dividers a space equal to the spaces upon the circumference of the plan, and placing one foot of the dividers at the point n. swing the other foot around till it cuts the arc drawn from point 2; then A n 2 will be the first triangle forming part of the envelope or pattern. With the same space between the points of the dividers, and 2 of the pattern as center, swing the dividers around again, cutting the arc drawn from point 3. Repeat this operation from 3 as center, or, in other words, continue to step from one arc to the next, until all the ares have been reached, as at g. which in this case will constitute one-half the pattern; after which, if desirable, the operation of stepping from arc to arc may be continued, as shown, finally reaching the point d. Draw d A and trace a line through the points obtained upon the arcs, as shown by n g d, which will complete the pattern.

Fig. 529   Pattern of Cant Shown in Fig. 527.

Fig. 529 - Pattern of Cant Shown in Fig. 527.