A crofton style formula and its application on the unit sphere
Yuanjun Chai, Young-Soo Lee · Bulletin of the Korean Mathematical Society · 1996
Crofton's formula on Euclidean plane $E^2$ states: Let $\Gamma$ be a rectifiable curve of length L and let G be a straight line. Then $$ \int_{G \cap \Gamma eq \phi} n dG = 2L $$ where n is the number of the intersection points of G with the curve $\Gamma$.