Fig. 265. - One-half Pattern of Side of Article Shown in Fig. 261.

Still another class of forms demanding treatment by triangulation result from the construction of arches

Principles of Pattern Cutting, out through curved walls, as when an arch of either round or elliptical form, as a door or window head, is placed in a circular wall in such a manner that its sides or jambs are radial, or fend toward the center of the curve of the wall, It will be seen that the soffit of such an arch is similar in shape to the sides of a transition piece, having what might be called its upper and lower surfaces curved and placed vertically. In such cases it is best, to consider the horizontal plane passing through the springing line of the arch as the base from which to measure the hights of all points assumed in the outer and inner curves.

It is believed that a sufficient number of this general class of problems will be f6und in the third section of the chapter on Pattern Problems to enable the careful student to apply the principles here explained to any new forms that might present themselves for his consideration, remembering that any form may be so turned as to bring any desired side into a horizontal position to be used as a base, or that an upper horizontal surface can be used as a base as well as a lower. The operations of triangulation undoubtedly require more care for the sake of accuracy than those of any other method of pattern cutting, for the reason that there is no opportunity of stepping off a continuous stretchout, at once, upon any line, either straight or curved. It is therefore not to be recommended if the subject in hand admits of treatment by any regular method without too much subdivision. Triangulation is not introduced as an alternate; method, but as a last resort, when nothing else will do.

Fig. 266 - Elevations and Plan of an Elliptical Cone.

Basides the various forms of transition pieces, another class of forms is to be treated under this head, which might almost be considered as regular tapering articles. They include shapes, or frustums cut from shapes, which terminate in an apex, but whose bases cannot be inscribed in a circle, as irregular polygons, figures composed of irregular curves as well as the perfect ellipse. A solid whose base is a perfect ellipse and whose apex is located directly over the center of its base (in other words, an elliptical cone) is perhaps the best typical representative of this class of figures. If the base of such a cone be divided into quarters by its major and minor axes, it will be seen at once that all of the points in the perimeter of any one quarter will be at different distances from the apex of the cone, because they are at different distances from the center of base or the intersection of the two axes. This is clearly shown in Fig. 266, in which are shown the two elevations and the plan of an elliptical cone. The side elevation shows K E to be the distance of the apex from the point P in the plan of the base, while the end elevation shows K' D to be the distance of the apex from the point D of the base, or the true distance represented by X D of the plan.

If one-quarter of the plan of the base, as D P, be divided into any convenient number of equal spaces and lines be drawn to the center X, as shown, each line will represent the horizontal distance of a point in the perimeter from the apex; and if a section of the cone be constructed upon any one of these lines, as, for instance, line 4 X, or, in other words, if a right angle triangle be drawn, of which 4 X is the base and R K the altitude, the hypothenuse will be the true distance of the point 4 from the apex. Therefore, to ascertain the distances from the apex to the various points in the circumference of the base construct a simple diagram of triangles, as shown in Fig. 207, viz.: Erect any perpendicular line, as X M, equal in hight to R K of the elevation; from X, on a horizontal line X P, as a base, set off the various distances of the plan, X 1, X 2, X 3, etc., numbering each point, and from each point draw a line to M. These hypothenuses will then represent the distances of the various points in the perimeter of the base from the apex of the cone; or, in other words, the sides of a number of triangles forming the envelope of the cone, the bases of which triangles will be the spaces 1 2, 2 3, etc., upon the plan. As all of these triangles terminate at. a, common apex or center, instead of laying out each one separately to form a pattern, as in the ease of an article of the type shown in Fig. 261, the simplest method is as follows:

Fig. 267. - Diagram of Sections on the Radial Lines of the Plan in Fig. 266, to which is Added the Pattern of One-quarter of the Envelope.

From M, of Fig. 267, as a center, with radii corresponding to the distances from M to points on P X, as M 1, M 2, M 3, etc., describe arcs indefinitely, as shown to the left; then taking the space used in stepping off the plan between the points of the dividers, place one foot upon the arc drawn from point 8, as at D, and swing the other foot around till it cuts the arc drawn from point 7; from this intersection as a center swing it around again, cutting the arc from 6; or in other words, step from one arc to the next till one-quarter of the circumference has been completed.

As the spaces in the base are equal, it is clearly a matter of convenience whether this last operation is begun upon arc 8, stepping first to arc 7, then to arc 6, etc., or whether it is begun upon arc 1, stepping first to arc 2, then to 3, etc., till complete. A line traced through these points, as A D, will give the cut at the base of the envelope, and A D M will be the envelope of one-quarter of the cone.