This question is related to my bath towel, which I hang on a rope, so let's have fun (you can use your own towel to do this experiment in bath-o).
There is this rectangle with sides $a. The rectangle is bent along a line that passes through the center of the rectangle. At which angle $\alpha$ (i. e., angle $\angle RAU$ – see the picture) should we bend the rectangle in order to get the minimum area of crossing intersection?
It is obvious that if $|a-b| \gg 0$, then $\alpha\approx\pi/4$. Also, we can look at the pentagon NOPQR. In addition to that, for a certain $a/b$ we get the triangle ($a/b<0.8150237, \alpha =\pi/4$).
All in all, I am looking for a graph: $a/b$ as in terms of $\alpha$ and the area of the crossing intersection, which is $\displaystyle S = [2ab - (a^2+b^2)t + 2abt^2 - (a^2+b^2)t^3]/[4(1-t^2)]$, where $t=\tan \alpha$.
Any help is highly welcome.