Epicycloid

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

An epicycloid with n − 1 cusps is given by the parametric equations

<math> x(\theta) = \cos \theta + {1 \over n} \cos n \theta, <math>

<math> y(\theta) = \sin \theta + {1 \over n} \sin n \theta. <math>

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid.

An epicycloid and its evolute are similar.[1]

See also: cycloid, hypocycloid, deferent and epicycle.