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.

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:

{\text{length}}(\gamma )={\frac 14}\iint n_{\gamma }(\varphi ,p)\;d\varphi \;dp.

The differential form

d\varphi \wedge dp

is invariant under rigid motions, so it is a natural integration measure for speaking of an "average" number of intersections. The right-hand side in the Crofton formula is sometimes called the Favard length.^[1]

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.

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 Crofton formula generalizes to any Riemannian surface; the integral is then performed with the natural measure on the space of geodesics.

Applications

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

Between two nested, convex, closed curves, the inner one is shorter.
Barbier's theorem: a curve of constant width w has a perimeter of πw.
The isoperimetric inequality: among closed curves with a given perimeter, the circle gives the unique maximum area.
The convex hull of any bounded rectifiable closed curve C has perimeter at most the length of C, with equality only when C is already a convex curve

References

↑ Luis Santaló (1976), Integral geometry and geometric probability, Addison-Wesley

Tabachnikov, Serge (2005). Geometry and Billiards. AMS. pp. 36–40. ISBN 0-8218-3919-5.
Santalo, L. A. (1953). Introduction to Integral Geometry. pp. 12–13, 54. LCC QA641.S3.

External links

Cauchy–Crofton formula page, with demonstration applets

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