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In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called epicycle — which rolls without slipping around a fixed circle. It is a particular kind of roulette.## See also

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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:

- $x(theta)\; =\; r\; (k+1)\; left(cos\; theta\; -\; frac\{cos((k+1)theta)\}\{k+1\}\; right)$

- $y(theta)\; =\; r\; (k+1)\; left(sin\; theta\; -\; frac\{sin((k+1)theta)\}\{k+1\}\; right).$

If k is an integer, then the curve is closed, and has k cusps (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 fills the space between the larger circle and a circle of radius R+2r.

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid.

An epicycloid and its evolute are similar.

- Special cases: Cardioid, Nephroid
- Cycloid
- Hypocycloid
- Epitrochoid
- Hypotrochoid
- Spirograph
- Deferent and epicycle

- J. Dennis Lawrence (1972).
*A catalog of special plane curves*. Dover Publications.

- Epicycloid, MathWorld
- "Epicycloid" by Michael Ford, The Wolfram Demonstrations Project, 2007

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Last updated on Friday September 26, 2008 at 13:45:18 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Friday September 26, 2008 at 13:45:18 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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