By the nature of the problem the pipe resolves itself, with respect to its section or profile, into some regular polygon. In the illustration presented in Fig. 459 an octagonal form is employed, but any other regular shape may be used, and the patterns for it will be cut by the same rule as here explained. Let N L V be any semicircle around which an octagonal pipe is to be carried. Draw N V, passing through the center W. Through W draw the perpendicular L K indefinitely. Let. N R be the required diameter of the octagon. Immediately below and in line with N R construct the profile ABC D F H G B and project the points B and C back upon N R and complete the elevation by drawing the semicircles O U and P T.

Fig. 459. - Elevation and Section.

Fig. 460. - Pattern.

A Semicircular Pipe with Longitudinal Seams.

By inspection of the diagram it is evident that the pattern for the sections corresponding to O U T P in the elevation may be pricked directly from the drawing as it is now constructed, and that the patterns for the sections represented by E A and D F of the profile will be plain straight strips of the width of one side of the figure, as shown by either E A or D F, and in length corresponding to the length of the sweep of the elevation on the lines N L V and R X S, respectively. By virtue of the bevel or flare of the pieces N L V U T and R X S T P, as shown by A B and C D of the profile, each becomes one-half of the frustum of a right cone, with its apex above or below the point W. Therefore prolong C D of the profile until it cuts the center line L K of the elevation in the point M. Then M D and M C are the radii of the pieces corresponding to P T S R of the elevation. Also prolong the side A B, or, for greater convenience, its equivalent, E G, until it cuts the center line in the point M1 Then M1 G and M' E arc the radii of the pieces corresponding to N L V U O of the elevation. From M1 in Fig. 460 as center, using each of the several radii in turn, strike arcs indefinitely, as shown by N1 V1, O1 U1, P1 T1 and R1 S1. Step off the length N L V in the elevation, Fig. 457, and make N1 V1 of Fig. 458 equal to it. Draw N1 O1 and V1 U1 radial to M2. Then N1 V1 U1 O1 will constitute the pattern for the pieces N L V U O of the elevation. In like manner establish the length of P1 T1, and draw P1 R1 and T1 S1, also radial to the center, as shown. Then P1 T1 S1 R1 will be the pattern for the pieces P T S X R of the elevation.

This rule may be employed for carrying any polygonal shape around any curve which is the segment of a circle. The essential points to be observed arc the placing of the profile in correct relationship to the elevation and to the central line L K, after which prolong the oblique sides until they cut the central line, thus establishing the radii by which their patterns may be struck. In the case of elliptical curves, by resolving them into segments of circles and applying this rule to each segment, as though it were to be constructed alone and distinct from the others, no difficulty will be met in describing patterns by the principles here set forth. The several sections may be united so as to produce a pattern in one piece by joining them upon their radial lines. This principle is further explained in the pattern for the curved molding in an elliptical window cap in Problem 128.