I'm trying to find the math behind laying out piping cuts. For instance, when one piece of pipe tees (at $90$ degrees) into another piece of pipe, how do you cut the pipe? I'm trying to figure out the math of creating a flat template that I can wrap around a pipe to draw a line and then cut the pattern with a torch for both parts when intersecting.
How do you draw the pattern when it's not a $90$, but a $45$, $30$, and $60$ degree tee.
To make a $ 90^{\circ}$ bend one needs to cut the tube at half the angle i.e., $45^{\circ}.$ For an $L$ full section and for Tee half section.
An intersection of a pipe and plane is an ellipse. The development of a pipe dia $D$ cut by a plane making angle $\alpha$ to diametral plane is always a cosine curve.
That is when thin flexible truncated pipe is laid flat, an ellipse boundary assumes the shape of a cosine curve.
This is the development of a sectioned or truncated cylinder . The development curve is to be drawn on paper to make the template. Length is $\pi D$ and the cosine wave height is $D\tan \alpha$. The paper is wrapped around the tube, keeping that as guide to hand-saw at angle $\alpha$ with a hacksaw.
Powered mitre saws are for larger production volumes. Any desired angle can be mitred. You only need to clamp the tube at desired $\alpha$ and lower the running saw.