en>Charles Matthews: expand lead, untag

2014-01-31T11:14:27Z

expand lead, untag

← Older revision		Revision as of 13:14, 31 January 2014
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	~~Alyson~~ is ~~what my spouse loves~~ to ~~call me but I don~~'~~t like when individuals use my full name~~. ~~Some time ago he selected to reside~~ in ~~North Carolina~~ and ~~he doesn~~'~~t strategy on changing it~~. The ~~favorite hobby for him and his children is to are psychics~~ [~~http~~://~~www~~.~~prayerarmor~~.~~com/uncategorized/dont-know-which-kind~~-of~~-hobby-~~to~~-take-up-read-~~the~~-following-tips~~/ ~~real psychic~~] ([~~http:~~//~~jplusfn~~.~~gaplus~~.~~kr/xe/qna/78647 just click~~ the ~~next document~~]~~) perform lacross and he would by no means give it up~~. ~~Office supervising~~ is ~~where my primary income arrives from but I~~'~~ve usually needed my own business~~.~~<br><br>my homepage~~ :~~: online psychics~~ [[http://~~www~~.~~taehyuna~~.~~net~~/xe/~~?document_srl=78721 Suggested Internet~~ page]]		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 value\|expected]] number of times a "random" [[line (mathematics)\|line]] intersects it.

			==Statement==
			Suppose γ is a [[curve\|rectifiable plane curve]]. Given an oriented line ''l'', let ''n''<sub>γ</sub>(''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 (mathematics)\|origin]]. The Crofton formula expresses the [[length of an arc\|arc length]] of the curve γ 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 γ 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 π''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–Crofton formula.

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

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

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

en>David Eppstein: /* External links */ not stub

2012-09-02T01:01:56Z

External links: not stub

New page

Alyson is what my spouse loves to call me but I don't like when individuals use my full name. Some time ago he selected to reside in North Carolina and he doesn't strategy on changing it. The favorite hobby for him and his children is to are psychics [http://www.prayerarmor.com/uncategorized/dont-know-which-kind-of-hobby-to-take-up-read-the-following-tips/ real psychic] ([http://jplusfn.gaplus.kr/xe/qna/78647 just click the next document]) perform lacross and he would by no means give it up. Office supervising is where my primary income arrives from but I've usually needed my own business.<br><br>my homepage :: online psychics [[http://www.taehyuna.net/xe/?document_srl=78721 Suggested Internet page]]

Iterated monodromy group - Revision history

en>Charles Matthews: expand lead, untag

en>David Eppstein: /* External links */ not stub