en>Brirush: Another reference

2014-10-01T17:04:21Z

Another reference

en>Brirush: /* Polar decomposition */ Finished adding in some new stuff

2013-11-16T04:45:00Z

Polar decomposition: Finished adding in some new stuff

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[[Image:EpitrochoidOn3-generation.gif|thumb|500px|The red curve is an epicycloid traced as the small circle (radius {{nowrap|''r'' {{=}} 1)}} rolls around the outside of the large circle (radius {{nowrap|''R'' {{=}} 3)}}.]]
In [[geometry]], an '''epicycloid''' is a plane [[curve]] produced by tracing the path of a chosen point of a [[circle]] — called an ''[[Deferent and epicycle|epicycle]]'' — which rolls without slipping around a fixed circle. It is a particular kind of [[roulette (curve)|roulette]].

If the smaller circle has radius ''r'', and the larger circle has radius ''R'' = ''kr'', then the
[[parametric equations]] for the curve can be given by either:
:<math>x (\theta) = (R + r) \cos \theta - r \cos \left( \frac{R + r}{r} \theta \right)</math>
:<math>y (\theta) = (R + r) \sin \theta - r \sin \left( \frac{R + r}{r} \theta \right),</math>
or:
:<math>x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \,</math>
:<math>y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right). \,</math>

If ''k'' is an integer, then the curve is closed, and has ''k'' [[cusp (singularity)|cusp]]s (i.e., sharp corners, where the curve is not
[[differentiable]]).

If ''k'' is a [[rational number]], say ''k=p/q'' expressed in simplest terms, then the curve has ''p'' cusps.

If ''k'' is an [[irrational number]], then the curve never closes, and forms a [[dense set|dense subset]] of the space between the larger circle and a circle of radius ''R'' + 2''r''.

<gallery caption="Epicycloid examples">
Image:Epicycloid-1.svg| ''k'' = 1
Image:Epicycloid-2.svg| ''k'' = 2
Image:Epicycloid-3.svg| ''k'' = 3
Image:Epicycloid-4.svg| ''k'' = 4
Image:Epicycloid-2-1.svg| ''k'' = 2.1 = 21/10
Image:Epicycloid-3-8.svg| ''k'' = 3.8 = 19/5
Image:Epicycloid-5-5.svg| ''k'' = 5.5 = 11/2
Image:Epicycloid-7-2.svg| ''k'' = 7.2 = 36/5
</gallery>

The epicycloid is a special kind of [[epitrochoid]].

An epicycle with one cusp is a [[cardioid]].

An epicycloid and its [[evolute]] are [[Similarity (geometry)|similar]].[http://mathworld.wolfram.com/EpicycloidEvolute.html]

==Proof==
[[Image:epicycloid geometry.svg|right|400px]]
We assume that the position of <math>p</math> is what we want to solve, <math>\alpha</math> is the radian from the tangential point to the moving point <math>p</math>, and <math>\theta</math> is the radian from the starting point to the tangential point.

Since there is no sliding between the two cycles, then we have that
:<math>\ell_R=\ell_r</math>
By the definition of radian (which is the rate arc over radius), then we have that
:<math>\ell_R= \theta R, \ell_r=\alpha r</math>
From these two conditions, we get the identity
:<math>\theta R=\alpha r</math>
By calculating, we get the relation between <math>\alpha</math> and <math>\theta</math>, which is
:<math>\alpha =\frac{R}{r} \theta</math>

From the figure, we see the position of the point <math>p</math> clearly.
:<math> x=\left( R+r \right)\cos \theta -r\cos\left( \theta+\alpha \right) =\left( R+r \right)\cos \theta -r\cos\left( \frac{R+r}{r}\theta \right)</math>
:<math>y=\left( R+r \right)\sin \theta -r\sin\left( \theta+\alpha \right) =\left( R+r \right)\sin \theta -r\sin\left( \frac{R+r}{r}\theta \right)</math>

==See also==
* Special cases: [[Cardioid]], [[Nephroid]]
* [[List of periodic functions]]
* [[Cycloid]]
* [[Hypocycloid]]
* [[Epitrochoid]]
* [[Hypotrochoid]]
* [[Spirograph]]
* [[Deferent and epicycle]]
* [[Epicyclic gearing]]

==References==
* {{cite book| author=J. Dennis Lawrence| title=A catalog of special plane curves| publisher=Dover Publications| year=1972| isbn=0-486-60288-5| pages=161,168–170,175}}

==External links==
*[http://demonstrations.wolfram.com/Epicycloid/ Epicycloid], [[MathWorld]]
*"[http://demonstrations.wolfram.com/Epicycloid/ Epicycloid]" by Michael Ford, [[The Wolfram Demonstrations Project]], 2007
*{{MacTutor|class=Curves|id=Epicycloid|title=Epicycloid}}
*[http://sourceforge.net/p/geofun/wiki/Home/ Sipirograph -- GeoFun]
*[http://link.springer.com/article/10.1007%2Fs12045-013-0106-3/ Historical note on the application of the epicycloid to the form of Gear Teeth]
[[Category:Curves]]
[[Category:Algebraic curves]]

[[nl:Cycloïde#Epicycloïde]]

Operator theory - Revision history

en>Brirush: Another reference

en>Brirush: /* Polar decomposition */ Finished adding in some new stuff