Vienna Standard Mean Ocean Water

2013-12-02T20:05:54Z

69.181.70.104:

[[Image:HypotrochoidOutThreeFifths.gif|thumb|450px|
The red curve is a hypotrochoid drawn as the smaller black circle rolls around inside the larger blue circle (parameters are ''R'' = 5, ''r'' = 3, ''d'' = 5).]]
A '''hypotrochoid''' is a [[roulette (curve)|roulette]] traced by a point attached to a [[circle]] of [[radius]] ''r'' rolling around the inside of a fixed circle of radius ''R'', where the point is a [[distance]] ''d'' from the center of the interior circle.

The [[parametric equation]]s for a hypotrochoid are:{{fact|date=February 2012}}

:<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 the angle formed by the horizontal and the center of the rolling circle (note that these are not polar equations because <math>\theta</math> is not the polar angle).

Special cases include the hypocycloid with ''d'' = ''r'' and the [[ellipse]] with ''R'' = 2''r''.

[[Image:Ellipse as hypotrochoid.gif|right|400|thumb|The [[ellipse]] (drawn in red) may be expressed as a special case of the hypotrochoid, with R = 2r; here R = 10, r = 5, d = 1.]]

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

==See also==

* [[List of periodic functions]]
* [[Epitrochoid]]
* [[Cycloid]]
* [[Hypocycloid]]
* [[Epicycloid]]
* [[Spirograph]]

==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=165–168 }}

==External links==

*[http://www.mekanizmalar.com/hypocycloid.html Flash Animation of Hypocycloid]
*[http://xahlee.org/SpecialPlaneCurves_dir/Hypotrochoid_dir/hypotrochoid.html Hypotrochoid] from Visual Dictionary of Special Plane Curves, Xah Lee
*[http://www.geogebra.org/de/upload/files/dynamische_arbeitsblaetter/yiorisos/g10.html Interactive hypotrochoide animation]
*{{MacTutor|class=Curves|id=Hypotrochoid|title=Hypotrochoid}}
{{Geometry-stub}}

[[Category:Curves]]

[[de:Zykloide#Epi- und Hypozykloide]]
[[ja:トロコイド#内トロコイド]]

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Vienna Standard Mean Ocean Water