This article describes Frank O’Rourke’s fourth rule of triangulation. This rule is “Symmetrical fittings having planes that are not parallel should be developed by using the elevation view without the plan view.” This elevation method is also called the short or simplified method.
There are many different layouts to describe this type of fitting; one that might work well is similar to one that Joseph J. Kaberlein described as the round taper with the mitered base. Although the fourth rule isn’t mentioned in his drawing, it’s apparently the case, as there is no “plan view” in his drawing.
The short method can be used to develop this fitting only if the cone is symmetrical in form, as shown in Figure 1. The cone must be developed with a symmetrical slant as Line AE and Line BF in Figure 1.
The first step to doing this is to draw an elevation view (your side view) of the cone to scale by drawing a line as shown from Point A to Point B in Figure 1. Bisect Line AB to find the center, and reference it as C.
Making points
Now using C as center, draw a line perpendicular to Line AB, shown as Line CD. Draw a line perpendicular to Line CD, establishing the top plane of the cone. Set your trammel points to the radius or one-half the diameter of the cone’s top. Now using Point D as center, make an arc from E to F that establishes points E and F.
Reset the trammel to the necessary bottom radius and using C as center, strike an arc from A to B, as shown. Draw your line from A to E and then from B to F. A symmetrical slant is now established - and now you may draw a line at any pitch that fits your application as shown from Point A to Point a in Figure 1. This does not change the slant of the cone.
The next thing to do is to establish the top and bottom profiles. The top profile is done by setting the trammel points from D to E and using D as center strike an arc from E to F. To draw the bottom profile, you must first establish the new center, Point C, by bisecting Line AB as shown in Figure 2.
Using C as center, set the trammels from C to A and strike an arc from A to B.
Once the profiles are established, divide both top and bottom profiles into equal sections. It’s important to understand that the top and bottom profiles must have the same number of sections in each. If you divide the top into six equal sections - Nos. 2, 4, 6, 8, 10, 12 and 14 - you must also divide the bottom profile into six equal sections shown as Nos. 1, 3, 5, 7, 9, 11 and 13 in Figure 3.
From each division along the profile, draw a line perpendicular to the plane, as shown in Figure 3, as 3’, 5’, 7’, 9’ and 11’ also as 4’, 6’, 8’, 10’ and 12’.
Slants
Now the slant of the fitting, the pitch of the bottom opening, the top and bottom profiles are established and you have divided both profiles into equal sections. You will use all this information to establish the true-length lines of the pattern. The idea of the “simplified method” of developing this pattern is to not have to make extra drawings that use the plan and elevation views together to generate a true-length diagram.
In Figure 4, you see the necessary true-length line diagram for each line as seen on the elevation drawing. To keep a clearer description, you split up the bottom profile into two sections. Let’s review section No. 1. Section No. 2 is included in Figure 4a and must be done in the same manner: Lines 1-2 and 13-14 in the elevation drawing are shown in true length.
Create two sides of a triangle (the height and base) to see the true-length line. The first step is to determine the height of each line on the elevation view. Begin this step by extending the plane of the bottom profile as shown, from X to Y in section No. 1 of Figure 4. Using 3’ as center and the trammels set from 3’ to 2, strike an arc from Point 2 of the top profile to Line XY and reference this as Line 3-2.
Reset the trammel points from 3’ to 4’ and again using Point 3’ as center, make an arc from Point 4’ to Line XY and reference this intersection as 3-4. You will need to continue this step with 5’, 7’, 9’, 11’ and 13.
Triangles
The second side or base of each triangle needs to be determined to find its true length. The length of each base is the difference of the sectional lines in the top profile and their respective sectional lines in the bottom profile.
To find the true length, transfer the distance from Point 4’ to Point 4 in the top profile and using Point 3 on the bottom profile, strike a reference mark shown as Line 3-4 in Figure 4 section No. 1.
Draw a line from Point 3-4 where it intersects Line XY (the height of the triangle) to your reference mark 3-4 on Line 3’-3 of the bottom profile (the base of the triangle). This line is the true length of Line 3’-4’ on the elevation. Continue these steps with Line 5’-4’ and Line 5’-6’ of the elevation.
Understand that the base length of 5’ to 4’ is the difference between 4 to 4’ of the top profile and 5 to 5’ of the bottom profile.
Referenced on Line 5-5’, Line 5’-6’ of the elevation view uses the difference of Line 6’-6 of the top profile and 5-5’ on the bottom profile. Repeat these steps for each of the points along the plane, Line XY of the bottom profile, 5’-7 of section No. 1 and points 9’, 11’ and 13 of Figure 4a’s section No. 2.
To develop the true-length line for elevation Line 12’-13, you have to first draw a line perpendicular to the bottom plane, as shown in section No. 2 of Figure 4a. Transfer the distance of Point 12’ to Point 12 in the top profile to this line, shown as reference 12-13 in Figure 4a.
Drawings
For laying out the pattern using the true-length lines from sections 1 and 2 of Figure 4 and Figure 4a, reference and transfer the points as you develop the pattern. Begin by drawing a straight line from 1 to 2 as shown in the half pattern; the true length of 1 to 2 is already shown in its true length in the elevation view. You need to only transfer this distance to the pattern.
Setting your trammel points to the true length of Point 3 to Point 2, obtained from section No. 1 and using Point 2 of the half pattern, make an arc near Point 3. Using Point 1 of the half pattern as center and the distance from Point 1 to Point 3 in the bottom profile, make an arc intersecting the arc struck from Point 2; this establishes point 3 on the pattern.
Setting your trammel points to the true length of Line 3-4 obtained from section No. 1 and using Point 3 of the half pattern, draw an arc near Point 4. Using Point 2 of the half pattern as center and the distance from Point 2 to Point 4 in the top profile, make an arc intersecting the previous arc struck from Point 3. This establishes Point 4 on the pattern.
Continue these steps to complete the pattern. Do each step on both sides as you develop the pattern. If you’re developing the pattern for repeated use, you can lay out the half pattern, cut and trace the opposite sides. A completed pattern can be seen in Figure 5.
When doing the simplified method, Line 3’-2 in the elevation has no height difference, so use the full length of Line 3’-3 of the bottom profile for the base length of Point 3’ to Point 2 in the elevation.
This is where you lose the accuracy that may be critical in some applications. In the traditional method, using the elevation view with a plan view, you will have the length of the base of Line 3’ to 2 to determine the actual true length of this line.
In this case the difference would have been only be a fraction of an inch, but this measurement will surly increase as the slant or pitch increases.
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