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

Statement

Suppose \gamma is a rectifiable plane curve. Given an oriented line ℓ, let n_\gamma(ℓ) be the number of points at which \gamma and ℓ intersect. We can parametrize the general line ℓ by the direction \varphi in which it points and its signed distance p from the origin. The Crofton formula expresses the arc length of the curve \gamma in terms of an integral over the space of all oriented lines: The differential form is invariant under rigid motions of \R^2, so it is a natural integration measure for speaking of an "average" number of intersections. It is usually called the kinematic measure. The right-hand side in the Crofton formula is sometimes called the Favard length. In general, the space of oriented lines in \R^n is the tangent bundle of S^{n-1}, and we can similarly define a kinematic measure on it, which is also invariant under rigid motions of \R^n. Then for any rectifiable surface S of codimension 1, we have where

Proof sketch

Both sides of the Crofton formula are additive over concatenation of curves, so it suffices to prove the formula for a single line segment. Since the right-hand side does not depend on the positioning of the line segment, it must equal some function of the segment's length. Because, again, the formula is additive over concatenation of line segments, the integral must be a constant times the length of the line segment. It remains only to determine the factor of 1/4; this is easily done by computing both sides when γ is the unit circle. The proof for the generalized version proceeds exactly as above.

Poincare’s formula for intersecting curves

Let E^2 be the Euclidean group on the plane. It can be parametrized as, such that each defines some : rotate by \varphi counterclockwise around the origin, then translate by (x, y). Then is invariant under action of E^2 on itself, thus we obtained a kinematic measure on E^2. Given rectifiable simple (no self-intersection) curves C, D in the plane, then The proof is done similarly as above. First note that both sides of the formula are additive in C, D, thus the formula is correct with an undetermined multiplicative constant. Then explicitly calculate this constant, using the simplest possible case: two circles of radius 1.

Other forms

The space of oriented lines is a double cover of the space of unoriented lines. The Crofton formula is often stated in terms of the corresponding density in the latter space, in which the numerical factor is not 1/4 but 1/2. Since a convex curve intersects almost every line either twice or not at all, the unoriented Crofton formula for convex curves can be stated without numerical factors: the measure of the set of straight lines which intersect a convex curve is equal to its length. The same formula (with the same multiplicative constants) apply for hyperbolic spaces and spherical spaces, when the kinematic measure is suitably scaled. The proof is essentially the same. The Crofton formula generalizes to any Riemannian surface or more generally to two-dimensional Finsler manifolds; the integral is then performed with the natural measure on the space of geodesics. More general forms exist, such as the kinematic formula of Chern.

Applications

Crofton's formula yields elegant proofs of the following results, among others: