Take $S^2$ with its standard metric. The space of great circles in $S^2$ can be identified with the real projective plane $\mathbb{R}P^2$. Let $X$ be an embedded circle in $S^2$; associate to it a function $f_X:\mathbb{R}P^2\rightarrow \mathbb{Z}\cup\{\infty\}$ which counts the number of intersection points (with multiplicity) of $X$ with given great circle. Can we reconstruct $X$ from $f_X$?

Remark: I think from some version of Crofton formula, it should be possible to determine the length of $X$ from $f_X$.