In my business we occasionally need to cut across large diameter plastic or steel pipe at a 45 degree angle. By wrapping a piece of thin carboard around a cut pipe I can make a template which, when layed flat, looks like a bell curve. Can you provide me with the general mathematical formula that defines this curve for any diameter pipe that we might have to cut?
Let's position the pipe in such a way as to have Z axis run along one of its generating straight lines, X axis pass through the center of its cross-section and Y axis tangent to the pipe (in the diagram on the left Y axis points away from the screen.) Assume also that the plane of the supposed cut is described by X+Z=2R where R is the radius of the pipe.
The top view is present in the diagram on the right. Let the central angle a be 0 at the bottom of the circle. The positive direction is taken counterclockwise, as usual. Then X=R-Rcos(a).
We want to express Z in terms of L - the arc length of the circumference. We can imagine unfolding of the latter to coincide with a segment of Y axis after being cut at the top point (2R, 0). Accordingly, let L change from 0 for a=-π to 2Rπ for a=π. This gives L(a)=Ra+Rπ. In other words a=L/R-π.
From here and X+Z=2R we get