Creating a miter-line intersection of a gore elbow and round cylinder is much easier than one might think.
The pattern can easily be made with a few simple parallel-line developments and projecting some lines.
As you draw your projection lines, establish the intersection of the two cylinders or pipes and demonstrate the developed patterns. Understand that the number sections in the profile will be the same amount of points that will define your intersection of the two pieces.
Once the profiles have been divided into equal sections as in Figure 1, draw horizontal projection lines from these points established on the profile (Profile 1) to the cylinder/pipe in the top view establishing points a, b, c and d on the cylinder, as shown in Figure 2a.
ProjectionsNow project lines from Profile 2 (the elevation view) through gore sections of the elbow (elevation view), remembering that lines drawn through each miter section must run parallel to the surface of the plane, as shown in Figure 2a. The intersection of the lines projected from the top view and the elevation view become the points of the miter line between the gore elbow and the cylinder, as shown in Figure 2b.
To establish each point correctly in Figure 2b, you must understand where your points come from. Look at Point a in the top view (Figure 2c). By looking at the two profiles, you can see projecting Point 7-1 over to the cylinder established this.
By looking at the elevation view (Figure 2c), you can see that points 7 and 1 are the top-center and bottom-center of the elbow. You also can see that Point a is the top- and bottom-center of the elbow and cylinder's miter line. You need to do this from each point projected from the profiles to establish the miter line points of the elbow and cylinder.
Once points a, b, c and d have been established as shown in the elevation view, Figure 2c, draw a line through these points establishing the miter line of the two partial gores for the elbow.
Before the partial gore patterns can be developed using parallel line development, you will still need to find the distance from 4 to x in Figure 2c. X represents the end of the two partial gores, which meet on the miter line as shown. From x, draw a line parallel to the other lines that were projected from the profile, as illustrated, and establish x on Profile 2. The distance from 4 to x on the profile is your true-length measurement for the parallel-line pattern you're about to develop.
With parallel-line development, you have a couple of choices on how to develop the stretch-out pattern: line projection and line transfer.
Line-projection methodStretching out the pattern of one gore using projection in parallel-line development is shown in Figure 3. Begin by extending two lines perpendicular to the gore (shown in red) and marking off the distances - the widths - of each section from the profile, starting with X to 4, 4 to 3, 3 to 2 and 2 to 1, and then back. Continue 1 to 2, 2 to 3, 3 to 4 and 4 to X. Extend the remaining points 1, 2, 3, 4, X, a, b, c and d. Make a reference mark on the corresponding line by following the two similar points to where they intersect. These are illustrated in Figure 3. A line extended from point 1 on the gore until it intersects Line 1 on the stretch-out is shown as 1'. Points 2, 3, and 4 are extended in the same fashion and are referenced at 2', 3' and 4'.
With extending a, b, c and d, you have to see what numbers they correspond to on the top view in Figure 2c or from the reference at the gore, a to 1, b to 2, c to 3 and d to 4. Connect the points and add your allowance for the seams (if any) and the pattern is ready.
Transferring line lengthsThe first step is to draw a reference line perpendicular to the gores plane as shown in Figure 4's Line za. It does not matter where the line is drawn. The illustration uses a as one end of the reference line because a is the start of the gore intersection to cylinder.
To begin this stretch-out, draw a line at random length as shown in the bottom image of Figure 4 as Line za. On the reference line, Line za, strike off the distance of each section from the profile until you have as many parallel spaces as sections. The illustration uses 12 sections.
Using your dividers, transfer all the points from the elevation view to your stretch-out view, as shown with Point 5. With one end on the reference line and the other on Point 5, the dividers were locked and this distance transferred to the stretch-out by placing one end on reference Line 5 and striking an arc across the perpendicular line, establishing Point 5. This is a very fast way to lay out these gores as well as many other fittings. No additional material for seams was allowed. Depending on the type of seam you're going to use, please remember to add this allowance to the gore sections.
Creating the opening in the cylinder/pipeThe cylinder has to be developed, or perhaps only the pattern for the correct opening. This too is done using the parallel-line development shown in Figure 5. Using the same line of intersection as on the cylinder, project all points perpendicular to the cylinder. From either the top or bottom line, make perpendicular lines to establish the six sections from the profile; the distance between each line would be the same as in the profile; points 7-1 to 6-2, etc. Using the corresponding points from the top view, Point d was established from Point 4 on the profile. Reference this where d and 4 intersect on the stretch-out. Continue to do this with the other points to create the pattern for the correct opening. If you're making the cylinder or pipe with the opening already established and cut out, you'll need to make the stretch-out the total length of the cylinder circumference.
More information on parallel lines is available in Wisconsin contractor Bud Goodman's November 2004 SNIPS article, "Line 'em up" at www.snipsmag.com and on Goodman's Web site, www.TheSheetMetalShop.com.