TITLE: Can we reconstruct the region in the xy plane by length measurements?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: A key search term is geometric tomography.
The problem posed is a version of Hammer’s X-ray reconstruction problem:

Gardner, Richard J., and Markus Kiderlen. "A solution to Hammer's X-ray reconstruction problem." Advances in Mathematics 214.1 (2007): 323-343.
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Two shapes with the same parallel X-rays.