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:

x(\theta )=(R+r)\cos \theta -r\cos \left({\frac {R+r}{r}}\theta \right)

y(\theta )=(R+r)\sin \theta -r\sin \left({\frac {R+r}{r}}\theta \right),

or:

x(\theta )=r(k+1)\cos \theta -r\cos \left((k+1)\theta \right)\,

y(\theta )=r(k+1)\sin \theta -r\sin \left((k+1)\theta \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 forms a dense subset of the space between the larger circle and a circle of radius R + 2r.

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.^[1]

Proof

We assume that the position of $p$ is what we want to solve, $\alpha$ is the radian from the tangential point to the moving point $p$ , and $\theta$ is the radian from the starting point to the tangential point.