Epicycloid
In geometry, an
epicycloid is a plane curve produced by tracing the path of a chosen point of a circle — called an
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 either: or: If
k is an integer, then the curve is closed, and has
k cusps. 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 forms a dense subset of the space between the larger circle and a circle of radius
R + 2
r. Epicycloid examples
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.