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In [[mathematics]], the '''Crofton formula''', named after [[Morgan Crofton]] (1826&ndash;1915), is a classic result of [[integral geometry]] relating the length of a curve to the [[expected value|expected]] number of times a "random" [[line (mathematics)|line]] intersects it.

==Statement==
Suppose &gamma; is a [[curve|rectifiable plane curve]]. Given an oriented line ''l'', let ''n''<sub>&gamma;</sub>(''l'') be the number of points at which &gamma; and ''l'' intersect. We can parametrize the general line ''l'' by the direction &phi; in which it points and its signed distance ''p'' from the [[origin (mathematics)|origin]]. The Crofton formula expresses the [[length of an arc|arc length]] of the curve &gamma; in terms of an [[integral]] over the space of all oriented lines:

:<math>\text{length} (\gamma) = \frac14\iint n_\gamma(\varphi, p)\; d\varphi\; dp.</math>

The [[differential form]]

:<math>d\varphi\wedge dp</math>

is invariant under [[Euclidean group|rigid motions]], so it is a natural integration measure for speaking of an "average" number of intersections.

==Proof sketch==
Both sides of the Crofton formula are [[additive function|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, and by additivity{{clarify|date=April 2012}} the function must be linear. It remains only to determine the factor of 1/4; this is easily done by computing both sides when &gamma; is the [[unit circle]].

==Other forms==
The space of oriented lines is a double [[covering map|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 everywhere|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 geometry|Riemannian]] surface; the integral is then performed with the natural measure on the space of [[geodesic]]s.

==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 &pi;''w''.
*The [[isoperimetric inequality]]: among closed curves with a given perimeter, the circle gives the unique maximum area.

== See also ==
* [[Buffon's noodle]]
* The [[Radon transform]] can be viewed as a measure-theoretic generalization of the Cauchy&ndash;Crofton formula.

==References==
*{{cite book | first=Serge | last=Tabachnikov | year=2005 | title=Geometry and Billiards | publisher=AMS | pages=36&ndash;40 | isbn=0-8218-3919-5}}
*{{cite book | first=L. A. | last=Santalo | year=1953 | title=Introduction to Integral Geometry | pages=12&ndash;13, 54 | id={{LCC|QA641.S3}}}}

== External links ==
* [http://merganser.math.gvsu.edu/david/reed03/projects/weyhaupt/project.html Cauchy&ndash;Crofton formula page], with demonstration applets

[[Category:Integral geometry]]
[[Category:Measure theory]]
[[Category:Differential geometry]]
[[Category:Theorems in geometry]]