Epicycloid
From Wikipedia, the free encyclopedia
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:
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.
 k = 1 |
 k = 2 |
 k = 3 |
 k = 4 |
 k = 2.1 = 21/10 |
 k = 3.8 = 19/5 |
 k = 5.5 = 11/2 |
 k = 7.2 = 36/5 |
The epicycloid is a special kind of epitrochoid.
An epicycle with one cusp is a cardioid.
An epicycloid and its evolute are similar.[1]
