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[[Image:Archimedean spiral.svg|right|thumb|300px|Three 360° turnings of one arm of an Archimedean spiral]]The '''Archimedean spiral''' (also known as the '''arithmetic spiral''') 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 that 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'' turns the spiral, while ''b'' controls the distance between successive turnings.

From the above equation, it can thus be stated: the position of particle from the point of start is proportional to the angle <math>\theta</math> as time elapses.

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

== Derivation of general equation of spiral ==
{{See also|Circular motion}}
A[[ Physics | physical approach ]] is used below to understand the notion of Archimedean spirals.
Suppose a point object moves in the [[Cartesian plane|cartesian system]] with a constant [[velocity]] <math>v</math> directed parallel to the x-axis, with respect to the x-y plane. Let at time <math>t=0</math>, the object was at an arbitary point <math>(c,0,0)</math>. If the x-y plane rotates with a constant [[angular velocity]] <math>\omega</math> about the Z-axis, then the velocity of the point with respect to Z-axis may be written as:   

<math>|v_0|=\sqrt{v^2+\omega^2(vt+c)^2}</math>   [[File:Spiral derivation...png|thumb|529x529px|The X-Y plane rotates to an angle ωt (anticlockwise) about the origin in time ''t''. ''(c,0)'' is the position of the object at t=0. ''P'' is the position of the object at time ''t'', at a distance of ''R=vt+c.''|alt=]]<math>v_x=v\ \cos (\omega t) - \omega\ (vt+c)\ \sin(\omega t)</math>
<math>v_y=v\ \sin{(\omega t)}+\omega \ (vt+c)\ \cos{(\omega t)}</math>

Here <math>vt+c</math> is the modulus of the [[Position (vector)|position vector]] of the particle at any time <math>t</math>,  <math>v_x</math> is the velocity component along x-axis and <math>v_y</math> is the component along y-axis. The figure shown alongside explains it.

<math>\int v_xdt =x</math><br><math>\int v_y dt = y</math>

The above equations can be integrated by applying [[integration by parts]], leading to the following parametric equations:

<math>x=(vt + c)\ \cos \omega t</math>        

<math>y=(vt+c)\ \sin \omega t</math>

Squaring the two equations and then adding (and some small alterations) results in the cartesian equation

<math>\sqrt{x^2+y^2}={v\over\omega}\cdot \arctan {y\over x} +c</math><br>

(using the fact that  <math>\omega t= \theta</math> and <math>\theta =\arctan {y\over x}</math>)

or

<math>\tan ((\sqrt{x^2+y^2}-c)\cdot {\omega \over v}) = {y\over x}</math>

Its polar form, 

<math>r= {v \over\omega} \cdot \theta +c</math>

==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]].[[File:Osculating circles of the Archimedean spiral.svg|thumb|right|[[Osculating circle|Osculating circles]] of the Archimedean spiral. The spiral itself is not drawn: we see it as the locus of points where the circles are especially close to each other.|207x207px]]

The Archimedean spiral has two arms, one for ''&theta;''&nbsp;>&nbsp;0 and one for ''&theta;''&nbsp;<&nbsp;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.

For large ''&theta;'' a point moves with well-approximated uniform acceleration along the Archimedean spiral while the spiral corresponds 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<ref>{{Cite OEIS|1=A091154}}</ref> (see contribution from Mikhail Gaichenkov).[[Image:Archimedean spiral polar.svg|right|thumb|300px|Archimedean spiral represented on a polar graph]]<br />

==General Archimedean spiral==

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

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

The normal Archimedean spiral occurs when ''c''&nbsp;=&nbsp;1. Other spirals falling into this group include the [[hyperbolic spiral]] (''c''&nbsp;=&nbsp;−1), [[Fermat's spiral]] (''c''&nbsp;=&nbsp;2), and the [[lituus (mathematics)|lituus]] (''c''&nbsp;=&nbsp;−2). 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]], due to Archimedes, makes use of an Archimedean spiral.  Archimedes also showed how the spiral can be used to [[angle trisection|trisect an angle]].  Both approaches relax the traditional limitations on the use of straightedge and compass in ancient Greek geometric proofs.<ref name=boyer>{{Cite book  | last = Boyer  | first = Carl B.  | title = A History of Mathematics  | publisher = Princeton University Press  | year = 1968 | location =Princeton, New Jersey  | pages = 140–142  | isbn = 0-691-02391-3}}</ref>

[[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 gases.<ref>{{Cite web|url=http://www.freepatentsonline.com/5603614.html|title=Fluid compressing device having coaxial spiral members|accessdate=2006-11-25|author1=Sakata, Hirotsugu  |author2=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 (although variable track spacing was later introduced to maximize the amount of music that could be cut onto a record).<ref>{{Cite web|url=http://ronpenndorf.com/journalofrecordedmusic5.html |title=Early Development of the LP |author=Penndorf, Ron |accessdate=2005-11-25 |archiveurl=https://web.archive.org/web/20051105045015/http://ronpenndorf.com/journalofrecordedmusic5.html |archivedate=5 November 2005 |url-status=dead }}. 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=https://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 journal|url=http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pubmed&pubmedid=4632851|title=Spiral Plate Method for Bacterial Determination|journal=Applied Microbiology|volume=25|issue=2|pages=244–52|author1=J. E. Gilchrist |author2=J. E. Campbell |author3=C. B. Donnelly |author4=J. T. Peeler |author5=J. M. Delaney |pmid=4632851|pmc=380780|year=1973}}</ref> They are also used to model the pattern that occurs in a roll of paper or tape of constant thickness wrapped around a cylinder.<ref name="uiuc">{{cite web|url=http://mtl.math.uiuc.edu/special_presentations/JoansPaperRollProblem.pdf |date=3 February 2009 |title=Joan's Paper Roll Problem |author=Tony Peressini |accessdate=2014-10-06 |url-status=dead |archiveurl=https://web.archive.org/web/20131103150639/http://mtl.math.uiuc.edu/special_presentations/JoansPaperRollProblem.pdf |archivedate=3 November 2013 }}</ref><ref name="google">{{cite book|title=Symmetry|author1=Walser, H.|author2=Hilton, P.|author3=Pedersen, J.|author4=Mathematical Association of America|date=2000|publisher=Mathematical Association of America|isbn=9780883855324|url=https://books.google.com/books?id=rU8jvsJMKsgC|page=27|accessdate=2014-10-06}}</ref>

==Code for producing an Archimedean spiral==
The following [[R programming|R]] code produces the first graph above. 
<syntaxhighlight lang="matlab">
a <- 1.5
b <- -2.4
t <- seq(0,5*pi, length.out=500)
x <- (a + b*t) * cos(t)
y <- (a + b*t) * sin(t)
plot(x,y, type="l", col=2, lwd=3)
abline(h=0, v=0, col="grey")
</syntaxhighlight>

==See also==
{{div col|colwidth=23em}}
* [[Archimedes' screw]]
* [[Euler spiral]]
* [[Fermat's spiral]]
* [[Golden spiral|Golden Spiral of Fibonacci]]
* [[Hyperbolic spiral]]
* [[List of spirals]]
* [[Logarithmic spiral]]
* [[Spiral of Theodorus]]
* [[Triple spiral|Triple spiral symbol]]
{{div col end}}

==References==
{{Reflist}}

==External links==
{{Commons category|Archimedean spirals}}
* [https://www.youtube.com/watch?v=SEiSloE1r-A Jonathan Matt making the Archimedean spiral interesting - Video : The surprising beauty of Mathematics] - [[TED Talks|TedX Talks]], [[Green Farms]]
* {{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)]
* [http://www.mathcurve.com/courbes2d.gb/archimede/archimede.shtml Archimedean spiral at "mathcurve"]

{{Spirals}}

{{DEFAULTSORT:Archimedean Spiral}}
[[Category:Squaring the circle]]
[[Category:Spirals]]
[[Category:Archimedes|Spiral]]