Editing Epitrochoid

{{short description|Plane curve formed by rolling a circle on the outside of another}}
[[File:EpitrochoidIn3.gif|thumb|400px|The epitrochoid with {{math|1=''R'' = 3}}, {{math|1=''r'' = 1}} and {{math|1=''d'' = 1/2}}]]

In [[geometry]], an '''epitrochoid''' ({{IPAc-en|ɛ|p|ᵻ|ˈ|t|r|ɒ|k|ɔɪ|d}} or {{IPAc-en|ɛ|p|ᵻ|ˈ|t|r|oʊ|k|ɔɪ|d}}) is a [[roulette (curve)|roulette]] traced by a point attached to a [[circle]] of [[radius]] {{mvar|r}} rolling around the outside of a fixed circle of radius {{mvar|R}}, where the point is at a distance {{mvar|d}} from the center of the exterior circle.

The [[parametric equation]]s for an epitrochoid are:

:<math>\begin{align}
& x (\theta) = (R + r)\cos\theta - d\cos\left({R + r \over r}\theta\right) \\
& y (\theta) = (R + r)\sin\theta - d\sin\left({R + r \over r}\theta\right)
\end{align}</math>
The parameter {{mvar|θ}} is geometrically the [[Polar coordinate system|polar angle]] of the center of the exterior circle. (However, {{mvar|θ}} is not the polar angle of the point <math>(x(\theta),y(\theta))</math> on the epitrochoid.)

Special cases include the [[limaçon]] with {{math|1=''R'' = ''r''}} and the [[epicycloid]] with {{math|1=''d'' = ''r''}}.

The classic [[Spirograph]] toy traces out epitrochoid and [[hypotrochoid]] curves.

The paths of planets in the once popular geocentric system of [[deferent and epicycle|deferents and epicycles]] are epitrochoids with <math>d>r,</math> for both the outer planets and the inner planets.

The orbit of the Moon, when centered around the Sun, approximates an epitrochoid.

The [[combustion chamber]] of the [[Wankel engine]] is an epitrochoid with {{math|1=''R'' = 2}}, {{math|1=''r'' = 1}} and {{math|1=''d'' = 1}}.

==See also==
* [[Cycloid]]
* [[Cyclogon]]
* [[Epicycloid]]
* [[Hypocycloid]]
* [[Hypotrochoid]]
* [[Spirograph]]
* [[List of periodic functions]]
* [[Rosetta (orbit)]]
* [[Apsidal precession]]

==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=[https://archive.org/details/catalogofspecial00lawr/page/160 160–164] | url-access=registration | url=https://archive.org/details/catalogofspecial00lawr/page/160 }}

==External links==
*[http://www.cs.aau.dk/~marius/gallery/generator.html Epitrochoid generator]
*{{MathWorld|Epitrochoid|Epitrochoid}}
*[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:Roulettes (curve)]]




[[ja:トロコイド#外トロコイド]]