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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 $f_X$.

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  • $\begingroup$ How exactly do you count intersections? With sign or without? How do you count tangency? $\endgroup$
    – Alexandre Eremenko
    Commented Oct 21, 2018 at 13:39
  • $\begingroup$ The relevant Crofton formula (computed with multiplicity, but without sign) is explained very clearly in Alvarez Paiva, Fernandez, 2007, Gelfand transforms and Crofton formulas: the length of a curve is its average number of intersections with circles, for a suitable constant multiple of the invariant density on the space of circles. $\endgroup$
    – Ben McKay
    Commented Oct 21, 2018 at 14:31

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This is related to the Funk transform. To a continuous function on the sphere it associates a map from the set of great circles to $\mathbb{R}$ that sends a great circle to the integral of the function along it. This transformation is invertible on even functions, that is one can reconstruct such a function from its integrals over the great circles.

(The Euclidean space analog is the Radon transform that was introduced at about the same time.)

In your case the function is not continuous, but a density concentrated on a curve. I think the inversion still works, but I am not completely sure.

On the other hand, if the curve is smooth (and generic in some way), then it should be sufficient to know those great circles which intersect it an odd number of times: the curve will be the envelope of this family.

As for the Crofton formula on the sphere, it exists indeed and is quite often used in differential geometry (for example in the proofs of Fenchel and Fary-Milnor theorems on the total curvature of space curves and knots).

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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.

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You can't distinguish a curve $C$ from the curve $-C$.

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    $\begingroup$ Yeah, I think the real question is reconstructing an embedded circle in $\mathbb{R}P^2$ from the data of its intersection multiplicities on the dual space of lines. $\endgroup$
    – Kevin Casto
    Commented Oct 21, 2018 at 7:02

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