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The famous Buffon's needle problem relates the area of a region $C$ (or length of a curve $C$, in 1D) to the expected number of times a randomly dropped needle will fall on the region.

This makes sense if the lengths of all needles are the same, and their distribution of location and angle is uniformly random. However, is there a generalization of the problem where the needle lengths, location and angle follow a given, nonuniform distribution?

I was thinking this could extend the problem to non-Euclidean settings - say instead of laying on $\mathbb{R}^n$, the region $C$ lays on a Riemannian manifold. In this case, could we come up with a Buffon's needle-type formula for the area of $C$ on the manifold? What would the distribution of needles be in this case?

blue_egg
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  • What have you tried? – David G. Stork Jul 26 '22 at 00:30
  • I don't see any mention of a finite region $C$ in the standard Buffon's problem. The problem is about the probability that the needle will intersect a set of parallel lines. – David K Jul 26 '22 at 00:50
  • @DavidK that is true! I suppose I'm asking about more general integral geometry formulas .. like cauchy-crofton, e.g., but I already asked a question about this – blue_egg Jul 27 '22 at 17:46
  • @DavidG.Stork nothing really, other than trying to find references. It looks like these types of results in integral geometry are focused on isometry-invariant measures, but I feel like there MUST be some result for a more general measure. – blue_egg Jul 27 '22 at 17:48

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