Epicycloid

File:EpitrochoidOn3-generation.gif

The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

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
Epicycloid-1.svg

k = 1
Epicycloid-2.svg

k = 2
Epicycloid-3.svg

k = 3
Epicycloid-4.svg

k = 4
Epicycloid-2-1.svg

k = 2.1 = 21/10
Epicycloid-3-8.svg

k = 3.8 = 19/5
Epicycloid-5-5.svg

k = 5.5 = 11/2
Epicycloid-7-2.svg

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

File:Epizykloide herleitung.svg

sketch for 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.