Imagine drawing a straight line $l$ through the center of a square piece of paper with area $1$. Now fold the paper along that line.
Q: What is the function for the area covered by the folded piece of paper $A_f(\theta)$, where $\theta$ denotes the orientation of $l$?
This is not a textbook problem, so I don't know how difficult it will be to come up with a solution, but here are my thoughts so far:
The function must have minima when all points of the folded part of the paper are projected on top of all points of the unfolded part, i.e. when $A_f(\theta=0,\frac{\pi}{4},\frac{\pi}{2},\frac{3}{4}\pi,...)=\frac{1}{2}$.
Because of the symmetry, we can restrict our investigation to $\theta \in[0,\frac{\pi}{4}]$.
Here is my strategy:
$1.$ Find the two sets of points, $S_l$ and $S_r$ ("left" and "right" of $l$, respectively), that $l$ divides the square $S$ into, as a function of $\theta$.
$2.$ Reflect $S_l$ around $l$, $S_l \mapsto S_l^*$.
$3.$ Find the area of points that make up the union $S_l^* \cup S_r$.
$1)$ $l=\left(\begin{smallmatrix}x\\y\end{smallmatrix}\right)=t\left( \begin{smallmatrix}1\\ \tan\theta\end{smallmatrix}\right)$ so $S_l=\{x,y : |x| \leq \frac{1}{2} \wedge y> x\tan\theta \}$ and $S_r=S \setminus S_l$.
$2)$ I have found the following formula for reflection, $\mathrm{Ref}_l(v) = 2\frac{v\cdot l}{|l|^2}l - v,$ where $v$ is being reflected around $l$. Ideally I would like to replace $v$ with $S_l$, but I've never tried to handle sets this way and don't really know how to proceed from here.
$3)$ Again, I'm not familiar with how this is done (or indeed if it can be done!). It seems to me that there's a long way from the abstract sets to the geometrical figures.
It is very possible that this can be done in a much simpler way; either way, I'd appreciate both alternative methods of solution as well as insights into how my strategy could be executed properly!
Thanks!