Create cutout template for vent pipe in 45 degree roof?

Question

How do I plot a roof cutout for a vertical pipe intersecting a 12/12 roof? Is there a formula for this projection, or a free software tool? I did cross section volumes of rotated parabolas with triple integrals in Calc III, but I can't figure out this simple plot.

Answer 1

The cutout is an ellipse.  The minor diameter is d, the diameter of the pipe.

If you look at the roof line from the side, you'll see the roof intersect the pipe at 45 degrees.  If the pipe's diameter is d, then the length of the ellipse's major diameter is h=d/cos(45).