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[[File:EpitrochoidIn3.gif|thumb|500px|The epitrochoid with ''R'' = 3, ''r'' = 1 and ''d'' = 1/2]]
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An '''epitrochoid''' ({{IPAc-en|ɛ|p|ɨ|ˈ|t|r|ɒ|k|ɔɪ|d}} or {{IPAc-en|ɛ|p|ɨ|ˈ|t|r|oʊ|k|ɔɪ|d}}) is a [[roulette (curve)|roulette]] traced by a point attached to a [[circle]] of [[radius]] ''r'' rolling around the outside of a fixed circle of radius ''R'', where the point is a distance ''d'' from the center of the exterior circle.
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The [[parametric equation]]s for an epitrochoid are
 
:<math>x (\theta) = (R + r)\cos\theta - d\cos\left({R + r \over r}\theta\right),\,</math>
:<math>y (\theta) = (R + r)\sin\theta - d\sin\left({R + r \over r}\theta\right).\,</math>
where <math>\theta</math> is a parameter (not the polar angle).
 
Special cases include the [[limaçon]] with ''R = r'' and the [[epicycloid]] with ''d = r''.
 
The classic [[Spirograph]] toy traces out epitrochoid and [[hypotrochoid]] curves.
 
The orbits of planets in the once popular geocentric [[Ptolemaic system]] are epitrochoids.
 
The [[combustion chamber]] of the [[Wankel engine]] is an epitrochoid.
 
==See also==
* [[Cycloid]]
* [[Epicycloid]]
* [[Hypocycloid]]
* [[Hypotrochoid]]
* [[Spirograph]]
* [[List of periodic functions]]
 
==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=160–164 }}
 
==External links==
*[http://www.cs.aau.dk/~marius/gallery/generator.html Epitrochoid generator]
*[http://mathworld.wolfram.com/Epitrochoid.html Epitrochoid at Mathworld]
*[http://xahlee.org/SpecialPlaneCurves_dir/Epitrochoid_dir/epitrochoid.html Visual Dictionary of Special Plane Curves on Xah Lee 李杀网]
*[http://gerdbreitenbach.de/planet/planet.html Interactive simulation of the geocentric graphical representation of planet paths ]
*{{MacTutor|class=Curves|id=Epitrochoid|title=Epitrochoid}}
*[http://sourceforge.net/p/geofun/wiki/Home/ Plot Epitrochoid -- GeoFun]
 
[[Category:Curves]]
 
 
{{Geometry-stub}}
 
[[ja:トロコイド#外トロコイド]]

Latest revision as of 15:04, 20 February 2014

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