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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?

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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. PDF download

          Xrays
          Two shapes with the same parallel X-rays.

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  • $\begingroup$ But to what extent does this answer the question? $\endgroup$
    – Todd Trimble
    Commented Aug 7, 2016 at 13:48
  • $\begingroup$ @ToddTrimble: Unless I misunderstand the question, one cannot reconstruct $S$ from the lengths of parallel ray intersections. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 7, 2016 at 13:50
  • $\begingroup$ My understanding is that we are supposed to consider lengths coming from multiple directions, not just the ones from segments parallel to the y-axis. $\endgroup$
    – Todd Trimble
    Commented Aug 7, 2016 at 13:52
  • $\begingroup$ @ToddTrimble: You are right; I misinterpreted. Then rotate the blue rays $90^\circ$. They yield the same length intersections. So two directions do not always suffice. $\endgroup$
    – Joseph O'Rourke
    Commented Aug 7, 2016 at 13:54
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    $\begingroup$ Okay, thanks, that's helpful! On the second page it seems to indicate that no set of three directions will suffice, but there are sets of four directions which will work for all curves with convex interior. Would that extend to all simple closed curves, convex or not? $\endgroup$
    – Todd Trimble
    Commented Aug 7, 2016 at 14:09

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