---
tag: 9D5NT00H6G
kind: statement
role: theorem
name: Crofton formula
msc: [53C65, 52A22, 28A75]
uses: [TAN5G5EQ96, EWS4WFTA7W, 0TH78S7GBZ, 0MP9BCF6DP, 06X5PF6YFP]
about: [0TH78S7GBZ, TAN5G5EQ96]
sources:
  - url: https://en.wikipedia.org/wiki/Crofton_formula
same_as:
  wikidata: Q3077613
  wikipedia: Crofton formula
---

## Statement

For $\theta \in [0, \pi)$ and $p \in \R$, let $L(\theta, p)$ be the [[line|0MP9BCF6DP]] $\set{x \in \R^2 : x_1 \cos \theta + x_2 \sin \theta = p}$.
Let $\gamma \colon [a, b] \to \R^2$ be a rectifiable [[plane curve|EWS4WFTA7W]] of [[length|0TH78S7GBZ]] $\ell(\gamma)$.
Let $n_\gamma(\theta, p) \in \set{0, 1, 2, \ldots} \cup \set{\infty}$ be the number of $t \in [a, b]$ with $\gamma(t) \in L(\theta, p)$.
Then $n_\gamma$ is measurable and
$$\ell(\gamma) = \frac{1}{2} \int_0^\pi \int_{-\infty}^{\infty} n_\gamma(\theta, p) \, dp \, d\theta.$$

## Remarks

Both sides are additive when the curve is cut into pieces, and both are invariant under rigid motions, since $dp \, d\theta$ is the invariant measure of [[integral geometry|TAN5G5EQ96]].
Take a segment of length $\ell$ in the direction $\varphi$, and fix $\theta$.
The lines $L(\theta, p)$ meeting it form an interval of $p$ of length $\ell \abs{\cos(\theta - \varphi)}$, and $\int_0^\pi \ell \abs{\cos(\theta - \varphi)} \, d\theta = 2 \ell$.
So the formula holds for polygonal curves, and the general case follows by approximating $\gamma$ by inscribed polygons and passing to the limit.
Example: a circle of radius $r$ meets $L(\theta, p)$ twice when $\abs{p} < r$, so the right side is $\frac{1}{2} \cdot \pi \cdot 2r \cdot 2 = 2 \pi r$.

## Explanations

### L1 human

The length of a curve equals, up to a fixed factor, the average number of times a random line crosses it.
Long or wiggly curves are crossed more often, and the formula makes this exact.

### L2 human

For the boundary of a plane convex body, almost every line that meets the body crosses the boundary twice.
So the perimeter equals $\int_0^\pi w(\theta) \, d\theta$, where $w(\theta)$ is the width in the direction $\theta$.
This is Cauchy's formula, and it gives Barbier's theorem for curves of constant width.
The formula explains [[Buffon's needle problem|06X5PF6YFP]].
It is the model for Crofton formulas in higher dimensions, which recover intrinsic volumes of convex bodies from intersections with random affine subspaces.