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# Epicycloid

[ep-uh-sahy-kloid]
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.

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:

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

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.