In Fig. 558, let ABCD represent the frustum of an oblique cone, and T S R V U the cylinder that joins the same at the angle indicated. The view here given of the frustum is that of its vertical side, so that A D and B C are the outlines of the slanting sides. In Fig. 559 E F G H shows the plan of the frustum at its base and K I J G the plan of the top, from which the side elevation is projected at the left, D C being the base and A D the vertical side. The intersecting cylinder is indicated by F L M G, and its profile by N O P Q. The diameter of cylinder is the same as that of top of frustum.

Fig. 559. - Elevation, Plan and Sections of the Frustum of a Scalene Cone Intersected Obliquely by a Cylinder.

Fig. 558. - Front Elevation of the Frustum of a Scalene Cone Intersected Obliquely by a Cylinder.

Divide the profile NOPQ into any convenient number of equal parts, and from the points thus obtained carry lines parallel with G M, cutting I J G and F G of plan, as shown. As the points in the profile of the cylinder lie in four vertical planes, indicated by the lines 7, 8 6, 1 5, and 2 4, it will be necessary, before their intersection can be obtained, to construct four vertical sections through the cone upon the lines I F, e f, J g, and h i. The point 3 requires no section; it being flush with the vertical side of the cone, must intersect somewhere on the line A D. To obtain the desired sections divide A D of elevation into any convenient number of equal parts, and from the points thus obtained erect lines parallel to the base D C, cutting B C. From the points in B C carry lines parallel with A G, cutting the center line E G, as shown. With b" and d" as centers, strike the arcs G b' and G d', thus forming sections of the cone in plan corresponding with a b and c d of elevation. The four vertical sections above referred to are shown below the plan by I' F', e' f, J' g' and h' i'. To avoid a confusion of lines, the method of obtaining the shapes is shown separately in Figs. 560 to 563, in which the reference letters are the same as in Fig. 559.

Fig. 560.

Fig. 581.

Fig. 582.

Fig. 563.

Sections of the Frustum of Scalene Cone Corresponding to Divisions in Profile of Cylinder in Fig. 559.

To obtain the shape of section on line I F in Fig. 560, extend E G, as indicated by I' D', which make equal to A D of the elevation, with its points of division a and c. From the points in I' D' erect the perpendiculars a b, c d and D' F'. With the T square placed parallel with I' D',' drop lines from the points in I F, cutting similar lines drawn from I' D'. A line traced through the points of intersection, as shown by I' F', will give the required shape. The sections shown in Figs. 561, 562 and 563 are obtained in a similar manner.

Having obtained these sections of the cone by the above method, arrange them as shown below the plan in Fig. 559. An inspection of the plan and profile will show that a line drawn from O of profile will cut section I F, line 8 6 of profile will cut section ef, line N P of profile will cut section J g, line 2 4 will cut section h i, and a line from Q of profile will cut the vertical side represented by G.

In connection with the sections in Fig. 559 draw an elevation of cylinder, as shown by S R V U, opposite the end of which draw a profile, as indicated by N' Q' P' O', commencing the divisions at the point N '. From the several points in the profile N' O' P' Q' carry lines parallel with U V against the several profiles I' F', e' f', J' g and h' i' as described above and as indicated by the small figures 1 to 8. A line traced through these points of intersection will give the miter line. A duplicate of this part of Fig. 559 is presented in Fig. 564 for the purpose of avoiding a confusion of lines. The miter line drawn through the intersecting lines is indicated by S T U.

Fig. 564. - Method of Obtaining Pattern of Cylinder Shown in Figs. 558 and 559.

Having now the profile of the cylinder and the miter line, all as shown, the pattern of the cylinder is obtained in accordance with the principles given in numerous examples in Section 1 of this Chapter, and as clearly shown in Fig. 564.

The method of obtaining the envelope of the frustum, and the opening in the side of the same to fit against the end of the cylinder just obtained, is shown in Fig. 565. The simple envelope of the frustum is obtained exactly as described in Problem 167, as will be seen by a comparison of Figs. 565 and 541. To obtain the opening in its side, however, involves an operation similar to that given in the problem immediately preceding. ABCD of Fig. 565 represents a side elevation of the frustum, as shown by the same letters in Fig. 559, and the vertical lines drawn through the same, designated by the small figures at the bottom, are the lines of the vertical sections obtained in Figs. 560-563, and correspond in numbers to the divisions in the profile in Fig. 559. To obtain the elevation of the opening, set off on each of these section lines the hights of the points of intersection occurring on corresponding sections as they appear in Fig. 564. Thus upon line designated at the bottom by 2 4, set off the vertical hights of points 2 and 4 on section h' i', Fig. 564, which section corresponds to line 2 4 of the profile, as shown in Fig. 559. In the same manner set off on line 1 5 the vertical hights of the points 1 and 5 on section J' g'. Obtain also points 8 and 6 from section e f and point 7 from section I' F', all as shown by the small figures. A line traced through these points will give the elevation of the opening. The outline of the opening has been shown in the plan, but its development is not necessary to the subsequent work of obtaining the pattern.

Divide the plan of the base of the frustum G F E into any convenient number of equal spaces, as indicated by the small letters. As an accurate elevation of the opening has now been obtained, this operation can be conducted without reference to any of the points previously used in obtaining the line of the opening. Therefore letters have been used in the divisions of G F E instead of figures so that no confusion may arise. From each of these points of division draw lines to G, which represents the plan of the apex of the cone. Also from so many of these points from which lines will cut the line of the opening, as a to f, erect lines vertically, cutting the base line D C, as shown by corresponding letters. From these points draw lines toward the apex of the cone X, cutting the line of the opening in the elevation, as shown but not lettered.

Proceed now to construct the diagram of triangles shown at the right, in which X1 D1 is equal to and parallel with X D, and in which A1 B1 and D1 C1 are drawn in continuation of A B and D C, as shown.

Fig. 565. - Method of Obtaining Opening in Side of Cone to Fit End of Cylinder.

Upon D1 C1, measuring from D1, set off the several lengths G b, G c, etc., of the plan, as shown by corresponding letters, and from the points thus obtained draw lines to X1, cutting the line A1 B1. From X1 as center draw arcs indefinitely from each of the points in D1 C1. From any convenient point upon arc a, as D2, draw a line to X1, which will form one side of the pattern of the desired envelope. Take between the points of the dividers a space equal to that used in dividing the plan G F E, and, placing one foot of the dividers at D2 step to arc b, thence to arc c, etc., till arc i is reached at G2 A line traced through these points will give the lower outline of the half of the envelope of the frustum which is pierced by the cylinder. From each of these points also draw lines toward X1, which intersect by arcs of corresponding number drawn with X1 as center from the line A1 B1, thus obtaining the upper line of the envelope.

From each of the points where the lines b b, c c, d d, etc., of the elevation cross the line of the opening project lines horizontally, cutting hypothenuses of corresponding letter in the diagram of triangles, all as shown by a1, b1 b2, c1 c2, etc. With one foot of the compasses at X', bring the pencil point successively to the points a1, b1, b2, etc., and draw arcs cutting radial lines in the pattern of corresponding letter. Then a line traced through the points thus obtained will be the required shape of the opening in the pattern.