en>CraigyDavi: Reverted 1 edit by 198.85.212.230 (talk).

2014-12-10T23:29:20Z

Reverted 1 edit by 198.85.212.230 (talk).

en>Alcides: /* Finding the radius of convergence */

2013-07-12T05:09:01Z

Finding the radius of convergence

← Older revision		Revision as of 07:09, 12 July 2013
Line 1:		Line 1:
	Hi, ~~everybody! My~~ name is ~~Inez~~. <br>It is a ~~little about myself~~: ~~I live~~ in ~~Denmark~~, ~~my city~~ of ~~Kobenhavn V~~. <br>~~It'~~s ~~called often Eastern or cultural capital~~ of ~~REGION SJALLAND~~. I'~~ve married 1 years ago~~.<br>~~I have two children -~~ a ~~son~~ (~~Randy~~) and the ~~daughter (Adrianne)~~. ~~We all like Seaglass collecting~~.<br><br>~~Review my weblog -~~ [http://www.~~freezehousebnb~~.~~com~~/~~louboutin~~.html ~~cheap christian louboutin shoes~~]		[[Image:Archimedean spiral.svg\|right\|thumb\|300px\|Three 360° turnings of one arm of an Archimedean spiral]]
			[[Image:Archimedean spiral polar.svg\|right\|thumb\|300px\|Archimedean spiral represented on a polar graph]]

			The '''Archimedean spiral''' (also known as the '''arithmetic spiral''' or '''spiral of Archimedes''') is a [[spiral]] named after the 3rd century BC [[Ancient Greece\|Greek]] [[mathematician]] [[Archimedes]]. It is the [[locus (mathematics)\|locus]] of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant [[angular velocity]]. Equivalently, in [[Polar coordinate system\|polar coordinates]] (''r'', ''θ'') it can be described by the equation

			:<math>\, r=a+b\theta</math>

			with [[real numbers]] ''a'' and ''b''. Changing the parameter ''a'' will turn the spiral, while ''b'' controls the distance between successive turnings.

			Archimedes described such a spiral in his book ''[[On Spirals]]''.

			==Characteristics==

			The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (equal to 2[[Pi\|π]]''b'' if ''θ'' is measured in [[radian]]s), hence the name "arithmetic spiral".

			In contrast to this, in a [[logarithmic spiral]] these distances, as well as the distances of the intersection points measured from the origin, form a [[geometric progression]].

			The Archimedean spiral has two arms, one for ''θ'' > 0 and one for ''θ'' < 0. The two arms are smoothly connected at the origin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the ''y''-axis will yield the other arm.

			===Separation distance between turns===

			Some sources describe the Archimedean spiral as a spiral with a "''constant separation distance''" between successive turns.<ref>"successive turnings of the Archimedean spiral have a constant separation distance" {{cite book
			\| last = Havil
			\| first = Julian
			\| title = Nonplussed! Mathematical Proof of Implausible Ideas
			\| publisher = Princeton Universoty Press
			\| location = Princeton, New Jersey
			\| year = 2007
			\| page = 109
			\| isbn = 978-0-691-12056-0}}
			</ref> This is somewhat misleading. The constant distances in the Archimedean spiral are measured along rays from the origin, which do not cross the curve at right angles, whereas a distance between [[parallel curve]]s is measured orthogonally to both curves. There is a curve slightly different from the Archimedean spiral, the [[Involute#Examples\|involute of a circle]], whose turns have constant separation distance in the latter sense of [[parallel curve]]s.

			==General Archimedean spiral==

			Sometimes the term ''Archimedean spiral'' is used for the more general group of spirals

			:<math>r=a+b\theta^{1\!/\!x}.</math>

			The normal Archimedean spiral occurs when ''x'' = 1. Other spirals falling into this group include the [[hyperbolic spiral]], [[Fermat's spiral]], and the [[lituus (mathematics)\|lituus]]. Virtually all static spirals appearing in nature are [[logarithmic spiral]]s, not Archimedean ones. Many dynamic spirals (such as the [[Parker spiral]] of the [[solar wind]], or the pattern made by a [[Catherine wheel (firework)\|Catherine's wheel]]) are Archimedean.

			==Applications==

			One method of [[squaring the circle]], by relaxing the strict limitations on the use of straightedge and compass in ancient Greek geometric proofs, makes use of an Archimedean spiral.

			[[Image:Two moving spirals scroll pump.gif\|frame\|Mechanism of a scroll compressor]]
			The Archimedean spiral has a variety of real-world applications. [[Scroll compressor]]s, made from two interleaved [[involute\|involutes of a circle]] of the same size that almost resemble Archimedean spirals, are used for compressing liquids and gases.<ref>{{Cite web\|url=http://www.freepatentsonline.com/5603614.html\|title=Fluid compressing device having coaxial spiral members\|accessdate=2006-11-25\|author=Sakata, Hirotsugu and Masayuki Okuda}}</ref> The coils of [[watch]] [[balance spring]]s and the grooves of very early [[gramophone record]]s form Archimedean spirals, making the grooves evenly spaced and maximizing the amount of music that could be fitted onto the record (although this was later changed to allow better sound quality).<ref>{{Cite web\|url=http://ronpenndorf.com/journalofrecordedmusic5.html\|title=Early Development of the LP\|author=Penndorf, Ron\|accessdate=2005-11-25\| archiveurl= http://web.archive.org/web/20051105045015/http://ronpenndorf.com/journalofrecordedmusic5.html\| archivedate= 5 November 2005 <!--DASHBot-->\| deadurl= no}}. See the passage on ''Variable Groove''.</ref> Asking for a patient to draw an Archimedean spiral is a way of quantifying human [[tremor]]; this information helps in diagnosing neurological diseases. Archimedean spirals are also used in [[digital light processing]] (DLP) projection systems to minimize the "[[rainbow effect]]", making it look as if multiple colors are displayed at the same time, when in reality red, green, and blue are being cycled extremely quickly.<ref>{{citation\|title=Handbook for Sound Engineers\|first=Glen\|last=Ballou\|publisher=CRC Press\|year=2008\|isbn=9780240809694\|page=1586\|url=http://books.google.com/books?id=zsEBavFYZuEC&pg=PA1586}}</ref> Additionally, Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter.<ref>{{Cite web\|url=http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pubmed&pubmedid=4632851\|title=Spiral Plate Method for Bacterial Determination\|author=J. E. Gilchrist, J. E. Campbell, C. B. Donnelly, J. T. Peeler, and J. M. Delaney}}</ref> They also are used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder.<ref>http://mtl.math.uiuc.edu/special_presentations/JoansPaperRollProblem.pdf</ref><ref>http://books.google.com.br/books?id=rU8jvsJMKsgC&pg=PA27&lpg=PA27&dq=paper+roll+thickness+archimedean+spiral&source=bl&ots=u7v-HK6_8M&sig=8UMxhOaMY8NM7XrS-2YEE1cTCTY&hl=pt-BR&sa=X&ei=ae_sTsDTJ8q2twf2woS-Cg&ved=0CE4Q6AEwBQ#v=onepage&q=paper%20roll%20thickness%20archimedean%20spiral&f=false</ref>

			==See also==
			* [[Hyperbolic spiral]]
			* [[Fermat's spiral]]
			* [[Logarithmic spiral]]
			* [[Triple spiral\|Triple spiral symbol]]
			* [[Fibonacci numbers\|Golden Spiral of Fibonacci]]

			==References==
			{{Reflist}}

			==External links==
			{{Commons category\|Archimedean spirals}}
			* {{MathWorld \| urlname=ArchimedesSpiral \| title=Archimedes' Spiral}}
			* {{PlanetMath \| urlname=ArchimedeanSpiral \| title=archimedean spiral \| id=5468}}
			* [http://www-groups.dcs.st-and.ac.uk/~history/Java/Spiral.html Page with Java application to interactively explore the Archimedean spiral and its related curves]
			* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Archimedean_spiral Online exploration using JSXGraph (JavaScript)]

			{{DEFAULTSORT:Archimedean Spiral}}
			[[Category:Spirals]]

en>Particle25: /* Theoretical radius */

2012-09-01T14:52:53Z

Theoretical radius

New page

Hi, everybody! My name is Inez. It is a little about myself: I live in Denmark, my city of Kobenhavn V. It's called often Eastern or cultural capital of REGION SJALLAND. I've married 1 years ago. I have two children - a son (Randy) and the daughter (Adrianne). We all like Seaglass collecting. Review my weblog - [http://www.freezehousebnb.com/louboutin.html cheap christian louboutin shoes]

Radius of convergence - Revision history

en>CraigyDavi: Reverted 1 edit by 198.85.212.230 (talk).

en>Alcides: /* Finding the radius of convergence */

en>Particle25: /* Theoretical radius */