The Cauchy-Crofton formula gives the Euclidean length of a curve as an expected number of intersections with random lines. Those random lines have to be distributed according to a measure that is invariant under rigid motions.
I read that a generalization of the C-C formula gives the Riemannian length of a curve under the same conditions; the measure then has to be invariant under the "natural measure on the space of geodesics." What does this mean?