Fig. 268. - Frustum of an Elliptical Cone.

In Fig. 268 is shown a perspective view of the frustum of the cone shown in Fig. 266, the upper surface A B being shown in Pig. 266 by the lines G H and N O. If the envelope of such a. frustum is desired the cut which its upper surface would make through the envelope of the entire cone could be obtained in exactly the same manner as that of its lower base, because the upper surface of the frustum is in reality the base of the cone, which remains above after the lower part has been cut away. Rut as part of the operation has already been performed in obtaining the cut at the base, it is most easily accomplished as follows: First draw radial lines from the point M of the diagram of triangles, Fig. 267, to each of the points previously obtained in the cut at the bottom of the envelope, between A and D; also draw a horizontal line a1 a hight above the base X P equal to R L, Fig. 266, cutting the hypothenuses M 1, M 2. etc., as shown by G H. Now place one foot of the dividers at the point M, and bringing the other foot successively to the various points of intersection of the line G H with the various hypothenuses, describe ares cutting the radial lines in the envelope of corresponding number. A line traced through the points of intersection, as B C, will give the cut at the top of the envelope of the frustum. of which A D is the bottom cut.

Fig. 269. - Elevation of the Frustum of an Elliptical Cone.

If the cut at the top of the frustum is to be oblique instead of horizontal, a means must be devised for measuring the distance from the apex at which the oblique plane cuts each of the hypothenuses, or in other words, each of the lines drawn from the apex of the cone to the various points in its base. In Fig. 269, E S T F is the elevation of an oblique frustum of an elliptical cone, whose apex is at K. and whose base is the same and has been divided in the same manner as that shown in Fig. 266.

Erect lines from each of the points in the curve of one-half the plan P D A to the base line E F of the elevation, thence carry them toward the apex K. cutting the line S T; the vertical hight of the points upon S T can then most easily be measured by carrying them horizontally, cutting the center line R K of the cone, where to avoid confusion they should be numbered to correspond with the points of the plan from which each was derived. These points may now be transferred in a body by any convenient means to the vertical line X' M' of the diagram of triangles, Fig. 270, seeing that each point is placed at the same distance from M' that it is from the point K of Fig. 269. A horizontal line from any one of the points on the line X' M' extended to the hypothenuse of corresponding number will then give the correct distance of that point from the apex of the cone. The diagram M' X' D' is a duplicate of M X P of Fig. 267, and the lower outline of the envelope is the same as that shown in Fig. 267. It will be noted, however, that half the stretchout of the base is necessary in this case to give all the essentials of the pattern of the envelope, while one-quarter was sufficient for the previous operations. When all the points in the uppen line of the frustum have been obtained in the diagram they may be transferred to the various radial lines in the envelope, from M' as a center, by the use of the compasses as before, all as shown in the drawing.

Fig. 270. - Diagram of Sections on the Radial Lines of the Plan in Fig. 269, with the Pattern of One-half the Envelope.

If the apex of the cone were not located directly over the crossing of the two axes of the ellipse - that is, if the cone were scalene or oblique instead of right - the met hod of obtaining its envelope, or parts of the same, would not differ from the foregoing. Lines drawn from the points of division in the circumference of the base to the point representing the position of the apex in the plan will be the horizontal distances used in constructing a diagram of triangles, which distances can be used in connection with the vertical hight of the cone, as before, in obtaining the various hypothenuses. If the apex of a scalene cone be located over the line of either axis of the ellipse, either within the perimeter of the base or upon one of those lines continued outside the base, one-half the pattern of the entire envelope will have to be obtained at one operation; but if the apex is not located upon either of those lines in the plan, then the entire envelope must be obtained at one operation, as no two quarters or halves of the cone will be exactly alike.

The method of obtaining the envelope of any scalene cone, even though its base be a perfect circle, is governed by the same principles as those employed in the above demonstrations.

It will be well to remember that any horizontal section of a scalene cone is the same shape as its base, which fact can be used to advantage in determining the best method to be employed in obtaining the envelope of any irregular flaring surface that may be presented. If, for instance, the plan of any article, whose upper and lower surfaces are horizontal, shows each to consist of two circles or parts of circles of different diameters not concentric, it is evident that the portion of the envelope indicated by the circles of the plan is part of the envelope of a scalene cone. An illustration of this is given in Fig. 271, which shows a portion of an article having rounded corners and flaring sides and ends, but with more flare at the end than at the side. The plan shows the curve of the bottom corner A B to be a quarter circle with its center at X, and that of the top C D to be a quarter circle with its center at Y. The rounded corner A B D C is then a portion of the envelope of a frustrum of a scalene cone, and the method of finding the dimensions of the complete cone is quite simple and is as follows: First draw a line, Z N, through the centers of the two circles in the plan, at right angles to which project an oblique elevation, as shown below, making the distance between the two lines E F and G H equal to the hight of the article. Lines from X and M of the plan of the bottom fall upon G H, locating the points X and H, while lines from Y and N of the top locate the points Y' and F in the upper line of the oblique elevation. A line drawn through Y' and X', the centers of the circles, will then represent the axis of the cone in elevation, which can be continued to meet a line drawn through the points F and H, representing the side of the cone, thus locating the apex Z' of the scalene cone. The point Z' can then be carried back to the plan, as shown at Z, thus locating the apex in that view. As the line N Z represents the horizontal distance between the point F and the apex Z' of the cone, so lines drawn from Z to any number of points assumed in the curve of the base C D will give the horizontal distances between those points and the apex, to be used as the bases in a diagram of triangles similar to that shown in Fig. 267, while V Z' gives their hight. Having drawn a diagram of triangles the pattern follows in the manner there shown.