Consider a closed smooth bounded curve enclosing a region S in the XY-plane. We know the function f(x), where x is a point on the x axis, defined as the length of the intersection of, the line paralell to the y-axis which goes through x, with the set S, so it can have multiple components and we know the total length. Suppose we know the same function for some coordiante systems rotated relative to the xy one. Can we reconstruct S using only this data? How many coordinate systems do we need?

Consider a closed smooth bounded curve enclosing a region $S$ in the XY-plane $\mathbb{R} ^2$.

We define the function $f(x)$, where $x$ is a point on the $x$ axis, as the length of the intersection of the line paralell to the $y$-axis which goes through $x$, with the set $S$, so it can have multiple components and we know the total length.

Suppose we know the same function for some coordiante systems rotated relative to the xy system. Can we reconstruct $S$ using only this data? How many coordinate systems do we need?