TITLE: Reconstructing a curve in $S^2$ from intersections with great circles
QUESTION [5 upvotes]: 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 $f_X$.

REPLY [2 votes]: The answer to this question, as stated, is "no". If infinitely many intersections are allowed, it is easy to construct plenty of different curves, each of them having infinitely many intersections with every great circle.