Edited to correct errors identified below:
I have a cylinder with r = 1 and a central axis given by y = (√3)x/6. I think this is a cylinder intersecting the xz-plane at π/6. I am pretty sure that the intersection is an ellipse, in which case the formula for the intersection of the cylinder with the xz-plane would be (x^2)/(a^2) + (z^2)/(b^2) = 1, and in this case, I think a=2, b=1. Please confirm that I have not blundered this too badly.
So far; so good.
Suppose I were to "mark" my cylinder with the intersection and unroll it into a plane. What is the curve created by the mark?
Here's why I'm interested in this question. I have to cut a piece of stove pipe with an eight inch radius such that the stove pipe will make a thirty degree angle with the adjacent surface. I can unroll the stove pipe into a flat sheet, but I need to draw the cut line on it.
Thanks for the help,
Chris.
Too long for a comment.
The cylinder equation is given by
$$ C(x,y,z) = \left(x+\frac{y}{2 \sqrt{3}}\right)^2-\frac{13}{12} \left(x^2+y^2+z^2-1\right)=0 $$
The intersection with the plane $x=0$ gives
$$ C_x(y,z) = \frac{12 y^2}{13}+z^2-1=0 $$
and with the plane $y=0$
$$ C_y(x,z) = \frac{x^2}{13}+z^2-1=0 $$