The Stove Pipe Problem 

What are the dimensions of the opening required to pass a cylindrical object through an inclined plane?

This is a trammel problem. The opening in the rake is elliptical in shape. The value for 1/2 the minor axis is already known, it is the same as the radius of the cylinder. See the drawing below for a method of determining the value for 1/2 the major axis.

1- Draw two intersecting lines, the angle, between which, is that of the rake. One line represents the slope of the rake, and the other represents the "horizontal."

2- From the intersection of the two lines draw an arc whose radius is that of the cylinder and which crosses the "horizontal.”

3- From the point where the arc crosses the "horizontal" erect a perpendicular that crosses the line representing the slope of the rake. The distance between the center of the arc and this point is the value of 1/2 the major axis.

A trammel constructed using these two values will describe the ellipse which must be cut in the rake to pass a cylinder of the stated radius. (see drawing below)

The same method may be use to create a gauge to mark out the cuts required to miter together two cylinders of the same radius (at any angle) where the diameter of the material is too great to allow it to be cut with available bench tools. 

Determine the angle at which the two cylinders want to be joined. 1/2 of this angle is the miter line. Use this angle as the slope of the rake in the above. The blank from which the ellipse has been removed can be used to mark out the necessary cuts. Care must be taken that the thickness of the gauge does not compromise the marked cut.