In some classes of sheet metal work certain forms arise for which patterns are required, but which cannot be classified under either of the two previous subdivisions. Their surfaces do not seem to be generated by any regular method. They are so formed that although perfectly straight lines can be drawn upon them (that is, lines running parallel with the form), such straight lines when drawn would not be parallel with each other; neither would they slant toward each other with any degree of regularity.

While in the systems described in the two previous portions of this chapter distances between lines running with the form measured at one end of an article govern those at the other end, in the forms considered in this department these distances are continually varying and bear no such relation to each other. Thus in parallel forms (moldings) the distance between any two lines running with the form is the same at both ends of the article, while in conical shapes all lines running with the form tend toward a common center or vertex, so that the distances between such lines at one end of the article (provided it does not reach to the vertex) bear a regular proportion to the distances between them at the other end. Hence, in the development of the pattern of an irregular form it becomes necessary to drop all previously described systems and simply proceed to measure up its surfaces, portion by portion, adding one portion to another till the entire surface has been covered.

To accomplish this end one of the most simple of all geometrical problems is made use of, to which the reader is referred (Chap. IV., Problem 36) - viz.: To construct a triangle, the lengths of the three sides being given.

As from any three given dimensions only one triangle can be constructed, this furnishes a correct means of measurement; and the solution of this problem in connection with a regular order and method of obtaining the lengths of the sides of the necessary triangles constitutes the entire system. To carry out this system it simply becomes necessary to divide the surface of any irregular object into triangles, ascertain the lengths of their sides from the drawing, and reproduce them in regular order in the pattern, and hence the term Triangulation is most fittingly applied to this method of development of surfaces.

In all articles whose sides lie in a vertical plane, distances can be measured in any direction across their sides upon an elevation of the article, but when the sides become rounded and slanting the length of a line running parallel with the form cannot be measured either upon the elevation or the plan. The elevation gives the distance from one end of the line to the other vertically or as it appears to slant to the right or left, but the distance of one end of the line forward or back of the other can only be obtained from the plan which while supplying this dimension does not give the hight. Consequently the true length of any straight line lying in the surface of any irregular form can only be ascertained by the construction of a right-angle triangle whose base is equal to the horizontal distance between the required points, and whose altitude is equal to the vertical distance of one point above the other, the hypothenuse giving the true distance between the points, or, in other words, the required length of the line.

For illustration. Fig. 261 shows an article which may be called a transition piece, the base of which,

Principled of Pattern Cutting.

A B C D of the plan, is a perfect circle lying in a borizontal plane, B H of the elevation. Its upper surface, however, N O P Q of the plan, is elliptical in shape and besides being placed at one side of the center is also in an inclined position, as shown by F G of the elevation. To the right of this plan is another drawing of the same, A' B' C' D', turned one-quarter around from which, and the elevation, is projected another view, J'K'LM, which may be called the front and which will assist in obtaining a more perfect conception of the shape of the article. A comparison of the three views shows that the slant of the sides is different at every point, and that the only dimensions of the article which can be measured directly upon the drawing are the circumference of the base and the slant hight, as given at E F, H G and L M.

Fig. 281.

Fig. 382.

Plans, Elevations, etc., of an Irregular Shaped Article, Illustrating the Principles of Triangulation.

Before a pattern of its side can be developed it will be necessary to ascertain its width (or distance from base to top) at frequent intervals and also its perimeter at the top. As F G, the distance across the top, is greater than N P (its apparent width in the plan), the curve N O P Q evidently does not give the correct distance around the top, and therefore a correct view of the top must be obtained. The method of accomplishing this does not differ from many similar operations described in. connection with parallel forms and is clearly shown in the drawing. Considering N O P Q as a correct plan or horizontal projection of the top, one-quarter of it, as O N, may be divided by any convenient number of points and their distances from N P set off upon the parallel lines drawn from N" P", thus obtaining O" N", one-quarter of the correct curve. It is more likely, however, that the correct shape of the top N" O" P" would be given, from which it would be necessary to obtain its correct appearance in the plan, which would be accomplished by drawing the normal curve in its correct relation to the line F G, as shown by N" O" P", when the raking process could be reversed, thereby developing the curve O N P one-half of the plan of top.