العربية
中国
Français
Deutsch
Ελληνική
हिन्दी
Italiano
日本語
Português
Русский
EspañolCROFTON FORMULA
(Redirected from Crofton\'s 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 γ is a rectifiable plane curve. Given an oriented line ''l'', let ''n''γ(''l'') be the number of points at which γ and ''l'' intersect. We can parametrize the general line ''l'' by the direction φ in which it points and its signed distance ''p'' from the origin. The Crofton formula expresses the arc length of the curve γ in terms of an integral over the space of all oriented lines:
:
The differential form
:
is invariant under rigid motions, so it is a natural integration measure for speaking of an "average" number of intersections.
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, and by additivity the function must be linear. 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 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.
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 π''w''.
★ The isoperimetric inequality: among closed curves with a given perimeter, the circle gives the unique maximum area.
★ Geometry and Billiards, , Serge, Tabachnikov, AMS, 2005, ISBN 0-8218-3919-5
★ {{cite book | first=L. A. | last=Santalo | year=1953 | title=Introduction to Integral Geometry | pages=12–13, 54 | id=
★ Cauchy-Crofton formula page, with demonstration applets
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.
| Contents |
| Statement |
| Proof sketch |
| Other forms |
| Applications |
| References |
| External links |
Statement
Suppose γ is a rectifiable plane curve. Given an oriented line ''l'', let ''n''γ(''l'') be the number of points at which γ and ''l'' intersect. We can parametrize the general line ''l'' by the direction φ in which it points and its signed distance ''p'' from the origin. The Crofton formula expresses the arc length of the curve γ in terms of an integral over the space of all oriented lines:
:
The differential form
:
is invariant under 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 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 the function must be linear. 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 π''w''.
★ The isoperimetric inequality: among closed curves with a given perimeter, the circle gives the unique maximum area.
References
★ Geometry and Billiards, , Serge, Tabachnikov, AMS, 2005, ISBN 0-8218-3919-5
★ {{cite book | first=L. A. | last=Santalo | year=1953 | title=Introduction to Integral Geometry | pages=12–13, 54 | id=
External links
★ Cauchy-Crofton formula page, with demonstration applets
This article provided by Wikipedia. To edit the contents of this article, click here for original source.
psst.. try this: add to faves
