As in the case of dividing the profile in parallel forms, this method is, theoretically, only approximate in accuracy, but the difference is so slight practically that it is not worth considering. Of course, the shorter the spaces are the greater is the accuracy. This method has, however, another significance which will be pointed out later on, which will help to simplify the solution of all tapering forms.
Fig. 248, - A Cone Truncated Obliquely.
If it is required that the cone should be truncated obliquely, as shewn in Fig. 248, it will be seen that all the points in the upper line of the frustum are at different distances from the base, or, what amounts to the same thing, from the apex of the original cone, hence some method of measuring these distances must be devised.
To explain the principles here involved more clearly, suppose that a cone be cut from a solid block of wood and of a hight and width to agree with some particular drawing, as, for instance, the one shown in Fig. 249. Divide the circle of the base E F upon the drawing into a convenient number of parts or spaces and mark the same number of points and spaces upon the edge of the base of the wooden cone, and from each of these points draw upon the sides of the wooden cone straight lines running to its apex.
Fig. 249. - Plan and Elevation from which to Construct a Wooden Cone for Purposes of Illustration.
A correct elevation of these lines upon the drawing may be obtained by carrying lines from the divisions or points in the plan of the base vertically till they strike the line of the base B C in the elevation, as shown in Fig. 250, thence to the apex A, cutting the line G H.
Now, if by means of a saw the upper part of the wooden cone be removed, being cut to the required angle as shown by the oblique line G H in the drawing, an opportunity is given, by the lines upon the part of the cone cut away, of measuring accurately the distance of each point of the curve thus produced from the apex.
Fig. 250. - Method of Obtaining the Lines upon the Elevation.
Then as all points in the base B C are equidistant from the apex A, to lay out the pattern of this frustum, first describe an arc of a circle whose radius is equal to the length of the side (or slant hight) of the cone A B, Fig. 250. Make this arc in length equal to the circumference of the base B C of the cone by means of the points, as previously described. To avoid confusion number these points 1, 2, 3, etc., from the starting point B, and from each of these points draw lines to the center of the arc, all as shown in Fig. 251.
Now, replacing that portion of the cone which was cut away so as to identify the lines upon its sides by the numbers at the base, the length of each line from the apex down to the cut can be measured by the dividers and transferred to the lines of the same numbers in the diagram, Fig. 251, as shown between G and H.
All this no doubt is quite simple when the model is at hand upon which to make the measurements. It is quite evident that it will not do to measure the distance upon the drawing. Fig. 250, from the apex A to the points of intersection on the line G H because the sides of the cone having an equal slant of flare all around, the lines upon the drawing do not represent the real distances except in the case of the two outside lines; the slant hight of a cone or any part of a cone being greater than the vertical hight of same part. But as these two outside lines do represent the correct slant of the cone on all sides, either one of them may be taken as a correct line upon which to measure these distances; that is, as a vertical section through the cone upon any or all of the lines drawn upon its sides. To make it a perfect section upon any one of these lines, say line 5, it is simply required that the position of the point of intersection of line 5 with the line G H be shown, which is done by carrying this point horizontally across till it strikes the side of the cone A B at 5, as illustrated in Fig. 250. The result of repeating this operation upon all the other lines is as though a thread or wire were stretched from the apex down along the side of the cone to the point B in the base and the cone were turned upon its axis, and as each line upon the side passes under the thread, the point where it cuts the intersecting plane G H where marked thereon, thus collecting all the points into one section as it were.
Fig. 2S1. - Method of Deriving the Pattern of a Frustum from the Wooden Model.
This operation is fully shown in Fig. 252, to which is added the development of the pattern, which is exactly the same as that shown in Fig. 251, the distances of the points between G and H from A being obtained in this case from the points upon the line A B, instead of from the model, as before. The points on A B are transferred to lines of corresponding number in the pattern by means of the compasses, as shown.
Should the frustum of which a pattern is required have both its upper and lower faces oblique to the axis of the cone a level base can be assumed at a convenient distance below the lower face of the frustum from which the circumference can lie obtained and then both the upper and lower faces of the frustum can be developed by the method just described.