I need to find the point were the curve crosses itself:
$\displaystyle x=2-\pi \cos(t)$
$\displaystyle y=2t-\pi \sin(t)$
This is really tricky; I can't think of any method to use to solve this (other than plotting points).
You need to find the parameter values s, t, that satisfy
$\displaystyle 2-\pi \cos t=2-\pi \cos s$ and $\displaystyle 2t-\pi\sin t=2s-\pi \sin s$
from the first equation you get that $\displaystyle \cos t=\cos s$, hence $\displaystyle s=\pm t+n\cdot 2\pi$.
Now plug that value of s into the second equation and hope for the best...