Crofton formula

In mathematics, the Crofton formula, named after Morgan Crofton (1826–1915), is a classic result of integral geometry relating the length of a curve to the expected number of times a "random" line intersects it.

Suppose ${\displaystyle \gamma }$ is a rectifiable plane curve. Given an oriented line l, let ${\displaystyle n_{\gamma }}$(l) be the number of points at which ${\displaystyle \gamma }$ and l intersect. We can parametrize the general line l by the direction ${\displaystyle \varphi }$ in which it points and its signed distance ${\displaystyle p}$ from the origin. The Crofton formula expresses the arc length of the curve ${\displaystyle \gamma }$ in terms of an integral over the space of all oriented lines:

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