This section is from the book "The New Metal Worker Pattern Book", by George Watson Kittredge. Also available from Amazon: The new metal worker pattern book.

In Fig. 315, let A B C be the section of the roof against which the flange is to fit, and let O P S R be the elevation of the pipe required to pass through the flange. Let the flange in size be required to extend from A to C over the ridge B. Since both sides of the roof are of the same pitch, both halves of the opening from the point B will be the same. Therefore, for convenience in obtaining both halves of the pattern at one operation, the line B C may be continued across the pipe toward A1, and used in place of B A, the distance from B on either line to the side O R being the same. Under these conditions it will be seen that the process of describing the pattern is identical with that in the previous problem. Make B A1 equal to B A, and proceed in the manner described in the problem just referred to. Divide the profile D E F G into any number of equal parts in the usual manner, and from the points so obtained carry lines vertically to the line A1 C, and thence, at right angles to it, indefinitely. Also carry lines in a similar manner from the points A1 and C. Draw H L parallel to A1 C. Make H I the width of the required flange, and draw I K parallel to H L. Through that part of the flange in which the center of the required opening is desired to be draw the line A1 C1, crossing the lines drawn from the profile. From each side of this line, on the several measuring lines, set off the same distance as shown upon the corresponding lines between D F and D E F, as shown. A line traced through the points thus obtained, as shown by D1 E1 F1 G1, will be the required opening to fit the pipe. Through the center, across the flange, draw the line N M, which represents the line of bend corresponding to the ridge B of the section of the roof.

Fig. 315. - The Pattern of a Flange to Fit Around a Pipe and Over the Ridge of a Roof.

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