Archimedean spiral 1

Archimedean spiral

Three 360° turnings of one arm of an Archimedean spiral

The Archimedean spiral (also known asthe arithmetic spiral) is a spiral namedafter the 3rd century BC Greekmathematician Archimedes. It is the locus ofpoints corresponding to the locations overtime of a point moving away from a fixedpoint with a constant speed along a linewhich rotates with constant angular velocity.Equivalently, in polar coordinates (r, θ) itcan be described by the equation

with real numbers a and b. Changing theparameter a will turn the spiral, while bcontrols the distance between successiveturnings.

Archimedes described such a spiral in hisbook On Spirals.

Characteristics

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral inpoints with a constant separation distance (equal to 2πb if θ is measured in radians), hence the name "arithmeticspiral".

In contrast to this, in a logarithmic spiral these distances, as well as the distances of the intersection points measuredfrom 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 theorigin. Only one arm is shown on the accompanying graph. Taking the mirror image of this arm across the y-axiswill yield the other arm.

Separation distance between turnsSome sources describe the Archimedean spiral as a spiral with a "constant separation distance" between successiveturns.[1] This is somewhat misleading. The constant distances in the Archimedean spiral are measured along raysfrom the origin which do not cross the curve at right angles, whereas a distance between parallel curves is measuredorthogonally to both curves. There is a curve slightly different from the Archimedean spiral, the involute of a circle,whose turns have constant separation distance in the latter sense of parallel curves.

Archimedean spiral 2

General Archimedean spiralSometimes the term Archimedean spiral is used for the more general group of spirals

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

ApplicationsOne method of squaring the circle, by relaxing the strict limitations on the use of straightedge and compass inancient Greek geometric proofs, makes use of an Archimedean spiral.

Mechanism of a scroll pump

The Archimedean spiral has a variety of real-world applications. Scrollcompressors, made from two interleaved Archimedean spirals of thesame size, are used for compressing liquids and gases.[2] The coils ofwatch balance springs and the grooves of very early gramophone recordsform Archimedean spirals, making the grooves evenly spaced andmaximizing the amount of music that could be fitted onto the record(although this was later changed to allow better sound quality).[3] Askingfor a patient to draw an Archimedean spiral is a way of quantifyinghuman tremor; this information helps in diagnosing neurologicaldiseases. Archimedean spirals are also used in digital light processing(DLP) projection systems to minimize the "rainbow effect", making itlook as if multiple colors are displayed at the same time, when in realityred, green, and blue are being cycled extremely quickly.[4] Additionally,Archimedean spirals are used in food microbiology to quantify bacterial concentration through a spiral platter.[5]

They also are used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around acylinder.[6][7]

References[1] "successive turnings of the Archimedean spiral have a constant separation distance" Havil, Julian (2007). Nonplussed! Mathematical Proof of

Implausible Ideas. Princeton, New Jersey: Princeton Universoty Press. p. 109. ISBN 978-0-691-12056-0.[2] Sakata, Hirotsugu and Masayuki Okuda. "Fluid compressing device having coaxial spiral members" (http:/ / www. freepatentsonline. com/

5603614. html). . Retrieved 2006-11-25.[3] Penndorf, Ron. "Early Development of the LP" (http:/ / ronpenndorf. com/ journalofrecordedmusic5. html). Archived (http:/ / web. archive.

org/ web/ 20051105045015/ http:/ / ronpenndorf. com/ journalofrecordedmusic5. html) from the original on 5 November 2005. . Retrieved2005-11-25.. See the passage on Variable Groove.

[4] Wilson, Tracy V.. "Adding Color and the Reliability of DLP" (http:/ / electronics. howstuffworks. com/ dlp1. htm). Archived (http:/ / web.archive. org/ web/ 20051217022540/ http:/ / electronics. howstuffworks. com/ dlp1. htm) from the original on 17 December 2005. . Retrieved2005-11-25.

[5] J. E. Gilchrist, J. E. Campbell, C. B. Donnelly, J. T. Peeler, and J. M. Delaney. "Spiral Plate Method for Bacterial Determination" (http:/ /www. pubmedcentral. nih. gov/ articlerender. fcgi?tool=pubmed& pubmedid=4632851). .

[6] http:/ / mtl. math. uiuc. edu/ special_presentations/ JoansPaperRollProblem. pdf[7] 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

Archimedean spiral 3

External links• Weisstein, Eric W., " Archimedes' Spiral (http:/ / mathworld. wolfram. com/ ArchimedesSpiral. html)" from

MathWorld.• archimedean spiral (http:/ / planetmath. org/ encyclopedia/ ArchimedeanSpiral. html) at PlanetMath• Page with Java application to interactively explore the Archimedean spiral and its related curves (http:/ /

www-groups. dcs. st-and. ac. uk/ ~history/ Java/ Spiral. html)• Online exploration using JSXGraph (JavaScript) (http:/ / jsxgraph. uni-bayreuth. de/ wiki/ index. php/

Archimedean_spiral)

Article Sources and Contributors 4

Article Sources and ContributorsArchimedean spiral  Source: http://en.wikipedia.org/w/index.php?oldid=522554677  Contributors: AnonMoos, AxelBoldt, Banus, Bryan Derksen, Cacycle, Charles Matthews, Chetvorno,Dannsuk, Discospinster, Doradus, Enyak, Everard Proudfoot, Flamingantichimp, Futurebird, Gandalf61, Giftlite, Gilliam, Goffrie, Graham87, Guyzero, Hannes Eder, Hmains, Incnis Mrsi,Iohannes Animosus, JMS Old Al, Jeronimo, Josh Grosse, Jsc83, Kelgann, Kindall, Kmhkmh, Konullu, LiDaobing, LittleDan, Luca Antonelli, Lzur, Mayooranathan, Mets501, Michael Hardy,Michael Kinyon, Motoball2, Noluz, Oekaki, OlEnglish, Oleg Alexandrov, Pbroks13, Pt, Qwyrxian, Radagast3, Rcgldr, Reddi, RexNL, Schneelocke, Seaphoto, Silly rabbit, Simetrical, SunCreator, Svick, The Man in Question, Torc2, VictorAnyakin, Willking1979, Wizard191, Xerxes314, Yalago, Ygrek, Zowie, 68 anonymous edits

Image Sources, Licenses and ContributorsImage:Archimedean spiral.svg  Source: http://en.wikipedia.org/w/index.php?title=File:Archimedean_spiral.svg  License: Creative Commons Attribution-Sharealike 2.5  Contributors: AdiJapan(talk)Image:Two moving spirals scroll pump.gif  Source: http://en.wikipedia.org/w/index.php?title=File:Two_moving_spirals_scroll_pump.gif  License: Creative Commons Attribution-ShareAlike3.0 Unported  Contributors: Cacycle

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