In Fig. 647, let a b c d be the elevation of the pipe, E E' its plan, A B C D the elevation of the flange and C D the angle or pitch of the roof. Since the projection of the base of the flange is required to be equal on all sides, as shown by C 1 and D 1, the flange will appear in the plan as a perfect circle, F F'. To avoid confusion of lines another elevation of the flange G H K J is shown in Fig. 648, below which is drawn a half plan of its base, M B N, and above which is a half plan of its top, G L H, all of which will be made use of in dividing the surface of the flange into measurable triangles for the purpose of developing a correct pattern of the same.

Divide the semicircle G L H into any convenient number of equal parts - in the present instance 12 - and from the points thus obtained drop perpendicular lines to G H. To obtain the shape of section on roof line J K divide the half plan of base M B N into the same number of equal parts as was G L H, and from the points thus obtained carry lines at right angles to M N, cutting J K. From the points in J K draw lines at right angles to it, as shown by a 1, b 2, c 3, etc. On these lines, measuring from J K, set off the length of corresponding lines in MNB, thus making lines a 1, b 2, c 3, etc., in J K C equal to lines a 1, b2, c 3, etc., in M N B. A line traced through these points, as shown by J C K, will give the shape of section on roof and furnish the stretchout of base for obtaining the pattern.

In Fig. 649 is drawn a duplicate of the plan in Fig. 647, the spaces in its outer line O D P being exact duplicates of the spaces in M B N of Fig. 648, and the spaces in its inner line O' D' P' being duplicates of those in G L H, all as shown by the small figures. Draw solid lines connecting similar points, as 1'1, 2' 2. 3' 3. etc. In like manner connect the points in O' D' P' with those of the next higher number in O D P, as O with 1', 1 with 2', 2 with 3', etc.. with dotted lines. These solid and dotted lines will then form the bases of a series of right angled triangles, whose altitudes can be derived from the elevation, and whose hypothenuses, when obtained, will be the correct measurements across the pattern between points of numbers corresponding with the lines across the plan.

Fig. 647. - Flan and Elevation of Pipe and Flange.

To construct the diagrams of triangles represented by solid and dotted lines in plan, extend G H of Fig. 648 indefinitely, as shown by II W. From the points in J K carry lines to the right indefinitely, parallel with G W, as shown by the lines between G W and K Y. At any convenient place, as R, and at right angles to GW, erect the line R S, cutting the base line K Y. From R set off the distance R T, equal to the length of any of the solid lines in plan, Fig. 649, as P' P, which is the horizontal distance between the pipe and lower edge of the flange. Draw T U parallel with R S, and also draw lines from the points in R S to T. For convenience the points in R S can be numbered to correspond with the points in J C K. Then the triangle T U S will correspond to a section through the article on the line P' P in plan, the hypothenuse S T representing the distance between the pipe and lower edge of the flange. The diagram of triangles in V W Y X is constructed in a similar manner; draw W Y at right angles to G W, and set off the space W V equal to the length of one of the dotted lines in plan, Fig. 649, as ft V, and draw lines from the points in W Y to V.

Fig. 649. - Plan of Flange, Showing Triangulation.

In developing the pattern the stretchout of top of flange where it joins the pipe can be obtained from the semicircle G L H. The stretchout of lower edge of flange where it joins the roof can be obtained from the section on the roof line J C K. The distance between points in O D P and O' D' P' of plan, Fig. 640, as indicated by the solid lines, is given in the diagram of triangles T R S. The distance between points as indicated by dotted lines in plan is given in the diagram V W Y. For the pattern then proceed as follows: Draw any line, as H' K', Fig. 650, equal in length to T S of first diagram of triangles. With the dividers set to the distance K 1 in K C .1 of section strike a small are (1') from the point 0' of pattern. With the dividers set to the distance V 1 of second diagram of triangles strike a small arc from the point 0 of pattern as center, cutting the first arc at 1' of pattern. From point 1' of pattern as center, and T 1 of first diagram of triangles as radius, describe a small arc (1), which intersect with one struck from 0 of pattern as center, and 0 1 in H L G as radius. Thus the points 0 0' and 1 1' of pattern are established. Proceed in this manner, using in the order described the stretchout obtained from the elliptical section K C J, the hy-pothenuse of triangle in second diagram corresponding to the dotted line in plan, the stretchout from the section G L H. and the hypothenuse of triangle in the first diagram corresponding to the solid line drawn across the plan, until all the measurements are used. Lines traced through the points thus obtained, as shown by H' G' and K' J', will be the half pattern. The other half of the pattern can be obtained by any means of duplication most convenient.

Fig. 648. - Elevation of Flange, with Plan, Sections and Diagrams of Triangles.

Fig. 650. - Half Pattern of Flange Shown in Fig. 647.

Should it be required to construct such a flange to fit over the ridge of a roof, it is clear that that part of the flange shown in the plan, Fig. 649, by O O' D'D would be a duplicate of the part shown by D' P' P D, and that, therefore, that portion of the pattern, Fig. 650, shown by 6 H' K' 6' would be one-quarter of the entire pattern, which could be duplicated so as to make either one-half or the whole pattern in one piece.