Editing Archimedean spiral

{{short description|Spiral with constant distance from itself}}
[[Image:Archimedean spiral.svg|right|thumb|300px|Three 360° loops of one arm of an Archimedean spiral]]
The '''Archimedean spiral''' (also known as '''Archimedes' spiral''', the '''arithmetic spiral''') is a [[spiral]] named after the 3rd-century BC [[Ancient Greece|Greek]] [[mathematician]] [[Archimedes]]. The term ''Archimedean spiral'' is sometimes used to refer to the more general class of spirals of this type (see below), in contrast to ''Archimedes' spiral'' (the specific arithmetic spiral of Archimedes). It is the [[locus (mathematics)|locus]] 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]] {{math|(''r'', ''θ'')}} it can be described by the equation
<math display=block>r = b\cdot\theta</math>
with [[real number]] {{mvar|b}}. Changing the parameter {{mvar|b}} controls the distance between loops.

From the above equation, it can thus be stated: position of the particle from point of start is proportional to angle {{mvar|θ}} as time elapses.

Archimedes described such a spiral in his book ''[[On Spirals]]''.  [[Conon of Samos]] was a friend of his and [[Pappus of Alexandria|Pappus]] states that this spiral was discovered by Conon.<ref>{{Cite book |last=Bulmer-Thomas |first=Ivor |title=[[Dictionary of Scientific Biography]] |volume=3 |page=391 |chapter=Conon of Samos |author-link=Ivor Bulmer-Thomas}}</ref>

== 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]] {{mvar|v}} directed parallel to the {{mvar|x}}-axis, with respect to the {{mvar|xy}}-plane. Let at time {{math|''t'' {{=}} 0}}, the object was at an arbitrary point {{math|(''c'', 0, 0)}}. If the {{mvar|xy}} plane rotates with a constant [[angular velocity]] {{mvar|ω}} about the {{mvar|z}}-axis, then the velocity of the point with respect to {{mvar|z}}-axis may be written as:

[[File:Spiral derivation...png|thumb|right|400px|The {{mvar|xy}} plane rotates to an angle {{mvar|ωt}} (anticlockwise) about the origin in time {{mvar|t}}. {{math|(''c'', 0)}} is the position of the object at {{math|''t'' {{=}} 0}}. {{mvar|P}} is the position of the object at time {{mvar|t}}, at a distance of {{math|''R'' {{=}} ''vt'' + ''c''}}.]]
<math display=block>\begin{align}
|v_0|&=\sqrt{v^2+\omega^2(vt+c)^2} \\
v_x&=v \cos \omega t - \omega (vt+c) \sin \omega t \\
v_y&=v \sin \omega t + \omega (vt+c) \cos \omega t
\end{align}</math>

As shown in the figure alongside, we have {{math|''vt'' + ''c''}} representing the modulus of the [[Position (vector)|position vector]] of the particle at any time {{mvar|t}}, with {{mvar|v<sub>x</sub>}} and {{mvar|v<sub>y</sub>}} as the velocity components along the x and y axes, respectively.

<math display="block">\begin{align}
\int v_x \,dt &=x \\
\int v_y \,dt &=y
\end{align}</math>

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

<math display=block>\begin{align}
x&=(vt + c) \cos \omega t \\
y&=(vt+c) \sin \omega t
\end{align}</math>

Squaring the two equations and then adding (and some small alterations) results in the Cartesian equation
<math display=block>\sqrt{x^2+y^2}=\frac{v}{\omega}\cdot \arctan \frac{y}{x} +c</math>
(using the fact that {{math|''ωt'' {{=}} ''θ''}} and {{math|''θ'' {{=}} arctan {{sfrac|''y''|''x''}}}}) or
<math display=block>\tan \left(\left(\sqrt{x^2+y^2}-c\right)\cdot\frac{\omega}{v}\right) = \frac{y}{x}</math>

Its polar form is
<math display=block>r= \frac{v}{\omega}\cdot \theta +c.</math>

== Arc length and curvature {{anchor|Arc length|Curvature}}==
[[File:Osculating circles of the Archimedean spiral.svg|thumb|upright=1.2|[[Osculating circle]]s of the Archimedean spiral, tangent to the spiral and having the same curvature at the tangent point. The spiral itself is not drawn, but can be seen as the points where the circles are especially close to each other.]]
Given the parametrization in cartesian coordinates
<math display=block>f\colon\theta\mapsto (r\,\cos \theta, r\,\sin \theta) = (b\, \theta\,\cos \theta,b\, \theta\,\sin\theta)</math>
the [[arc length]] from {{math|''θ''<sub>1</sub>}} to {{math|''θ''<sub>2</sub>}} is
<math display=block>\frac{b}{2}\left[\theta\,\sqrt{1+\theta^2}+\ln\left(\theta+\sqrt{1+\theta^2}\right)\right]_{\theta_1}^{\theta_2}</math>
or, equivalently:
<math display=block>\frac{b}{2}\left[\theta\,\sqrt{1+\theta^2}+\operatorname{arsinh}\theta\right]_{\theta_1}^{\theta_2}.</math>
The total length from {{math|''θ''<sub>1</sub> {{=}} 0}} to {{math|''θ''<sub>2</sub> {{=}} ''θ''}} is therefore
<math display=block>\frac{b}{2}\left[\theta\,\sqrt{1+\theta^2}+\ln \left(\theta+\sqrt{1+\theta^2} \right)\right].</math>

The curvature is given by 
<math display=block>\kappa=\frac{\theta^2+2}{b\left(\theta^2+1\right)^\frac{3}{2}}</math>

==Characteristics==
[[File:Archimedean spiral polar.svg|right|thumb|upright=1.2|Archimedean spiral represented on a polar graph]]
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 {{math|2''[[Pi|π]]b''}} if {{mvar|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 {{math|''θ'' > 0}} and one for {{math|''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 {{mvar|y}}-axis will yield the other arm.

For large {{mvar|θ}} 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).

As the Archimedean spiral grows, its [[evolute]] asymptotically approaches a circle with radius {{math|{{sfrac|{{abs|''v''}}|''ω''}}}}.

==General Archimedean spiral==

Sometimes the term ''Archimedean spiral'' is used for the more general group of spirals
<math display=block>r = a + b\cdot\theta^\frac{1}{c}.</math>

The normal Archimedean spiral occurs when {{math|''c'' {{=}} 1}}. Other spirals falling into this group include the [[hyperbolic spiral]] ({{math|''c'' {{=}} −1}}), [[Fermat's spiral]] ({{math|''c'' {{=}} 2}}), and the [[lituus (mathematics)|lituus]] ({{math|''c'' {{=}} −2}}).

==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>

[[File:Two moving spirals scroll pump.gif|thumb|upright=0.5|Mechanism of a scroll compressor]]
The Archimedean spiral has a variety of real-world applications. [[Scroll compressor]]s, used for compressing gases, have rotors that can be made from two interleaved Archimedean spirals, [[involute|involutes of a circle]] of the same size that almost resemble Archimedean spirals,<ref>{{Cite web |last=Sakata |first=Hirotsugu |last2=Okuda |first2=Masayuki |title=Fluid compressing device having coaxial spiral members |url=http://www.freepatentsonline.com/5603614.html |access-date=2006-11-25}}</ref> or hybrid curves.

Archimedean spirals can be found in [[spiral antenna]], which can be operated over a wide range of frequencies.

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 |last=Penndorf |first=Ron |title=Early Development of the LP |url=http://ronpenndorf.com/journalofrecordedmusic5.html |url-status=dead |archive-url=https://web.archive.org/web/20051105045015/http://ronpenndorf.com/journalofrecordedmusic5.html |archive-date=5 November 2005 |access-date=2005-11-25}}. 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 |last=Gilchrist |first=J. E. |last2=Campbell |first2=J. E. |last3=Donnelly |first3=C. B. |last4=Peeler |first4=J. T. |last5=Delaney |first5=J. M. |year=1973 |title=Spiral Plate Method for Bacterial Determination |journal=Applied Microbiology |volume=25 |issue=2 |pages=244–52 |doi=10.1128/AEM.25.2.244-252.1973 |pmc=380780 |pmid=4632851}}</ref>

[[File:Celestial spiral with a twist.jpg|thumb|upright|[[Atacama Large Millimeter Array]] image of [[LL Pegasi]]]]
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 |last=Peressini |first=Tony |date=3 February 2009 |title=Joan's Paper Roll Problem |url=http://mtl.math.uiuc.edu/special_presentations/JoansPaperRollProblem.pdf |url-status=dead |archive-url=https://web.archive.org/web/20131103150639/http://mtl.math.uiuc.edu/special_presentations/JoansPaperRollProblem.pdf |archive-date=3 November 2013 |access-date=2014-10-06}}</ref><ref name="google">{{Cite book |last=Walser |first=H. |url=https://archive.org/details/symmetry0000wals |title=Symmetry |last2=Hilton |first2=P. |last3=Pedersen |first3=J. |date=2000 |publisher=Mathematical Association of America |isbn=9780883855324 |page=[https://archive.org/details/symmetry0000wals/page/27 27] |access-date=2014-10-06 |url-access=registration}}</ref>

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. For instance, the star [[LL Pegasi]] shows an approximate Archimedean spiral in the dust clouds surrounding it, thought to be ejected matter from the star that has been shepherded into a spiral by another companion star as part of a double star system.<ref>{{cite journal
 | last1 = Kim | first1 = Hyosun
 | last2 = Trejo | first2 = Alfonso
 | last3 = Liu | first3 = Sheng-Yuan
 | last4 = Sahai | first4 = Raghvendra
 | last5 = Taam | first5 = Ronald E.
 | last6 = Morris | first6 = Mark R.
 | last7 = Hirano | first7 = Naomi
 | last8 = Hsieh | first8 = I-Ta
 | date = March 2017
 | doi = 10.1038/s41550-017-0060
 | issue = 3
 | journal = Nature Astronomy
 | title = The large-scale nebular pattern of a superwind binary in an eccentric orbit
 | volume = 1| page = 0060
 | arxiv = 1704.00449
 | bibcode = 2017NatAs...1E..60K
 | s2cid = 119433782
 }}</ref>

==Construction methods==

The Archimedean Spiral cannot be constructed precisely by traditional compass and straightedge methods, since the arithmetic spiral requires the radius of the curve to be incremented constantly as the angle at the origin is incremented. But an arithmetic spiral can be constructed approximately, to varying degrees of precision, by various manual drawing methods. One such method uses compass and straightedge; another method uses a modified string compass. 

The common traditional construction uses compass and straightedge to approximate the arithmetic spiral. First, a large circle is constructed and its circumference is subdivided by 12 diameters into 12 arcs (of 30 degrees each; see regular [[dodecagon]]). Next, the radius of this circle is itself subdivided into 12 unit segments (radial units), and a series of concentric circles is constructed, each with radius incremented by one radial unit. Starting with the horizontal diameter and the innermost concentric circle, the point is marked where its radius intersects its circumference; one then moves to the next concentric circle and to the next diameter (moving up to construct a counterclockwise spiral, or down for clockwise) to mark the next point. After all points have been marked, successive points are connected by a line approximating the arithmetic spiral (or by a smooth curve of some sort; see [[French Curve]]). Depending on the desired degree of precision, this method can be improved by increasing the size of the large outer circle, making more subdivisions of both its circumference and radius, increasing the number of concentric circles (see [[List of spirals|Polygonal Spiral]]). Approximating the Archimedean Spiral by this method is of course reminiscent of Archimedes’ famous method of approximating π by doubling the sides of successive polygons (see [[Pi#Polygon approximation era|Polygon approximation of π]]).

Compass and straightedge construction of the [[Spiral of Theodorus#Archimedean spiral|Spiral of Theodorus]] is another simple method to approximate the Archimedean Spiral.

A mechanical method for constructing the arithmetic spiral uses a modified string compass, where the string wraps and winds (or unwraps/unwinds) about a fixed central pin (that does not pivot), thereby incrementing (or decrementing) the length of the radius (string) as the angle changes (the string winds around the fixed pin which does not pivot). Such a method is a simple way to create an arithmetic spiral, arising naturally from use of a string compass with winding pin (not the loose pivot of a common string compass). The string compass drawing tool has various modifications and designs, and this construction method is reminiscent of string-based methods for creating ellipses (with two fixed pins).

Yet another mechanical method is a variant of the previous string compass method, providing greater precision and more flexibility. Instead of the central pin and string of the string compass, this device uses a non-rotating shaft (column) with helical threads (screw; see [[Archimedes' screw|Archimedes’ screw]]) to which are attached two slotted arms: one horizontal arm is affixed to (travels up) the screw threads of the vertical shaft at one end, and holds a drawing tool at the other end; another sloped arm is affixed at one end to the top of the screw shaft, and is joined by a pin loosely fitted in its slot to the slot of the horizontal arm. The two arms rotate together and work in consort to produce the arithmetic spiral: as the horizontal arm gradually climbs the screw, that arm’s slotted attachment to the sloped arm gradually shortens the drawing radius. The angle of the sloped arm remains constant throughout (traces a [[cone]]), and setting a different angle varies the pitch of the spiral. This device provides a high degree of precision, depending on the precision with which the device is machined (machining a precise helical screw thread is a related challenge). And of course the use of a screw shaft in this mechanism is reminiscent of [[Archimedes' screw|Archimedes’ screw]].

==See also==
{{Portal|Mathematics}}
<!-- alphabetical order please [[WP:SEEALSO]] -->
<!-- please add a short description [[WP:SEEALSO]], via {{subst:AnnotatedListOfLinks}} or {{Annotated link}} -->
{{div col|colwidth=20em|small=yes}}
* {{Annotated link |Archimedes' screw}}
* {{Annotated link |Fermat's spiral}}
* {{Annotated link |Golden spiral}}
* {{Annotated link |Hyperbolic spiral}}
* {{Annotated link |List of spirals}}
* {{Annotated link |Logarithmic spiral}}
* {{Annotated link |Spiral of Theodorus}}
* {{Annotated link |Triple spiral|Triple spiral symbol}}
{{div col end}}
<!-- alphabetical order please [[WP:SEEALSO]] -->

==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}}
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