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:

$x (\theta) = (R %2B r) \cos \theta - r \cos \left( \frac{R %2B r}{r} \theta \right)$

$y (\theta) = (R %2B r) \sin \theta - r \sin \left( \frac{R %2B r}{r} \theta \right),$

or:

$x (\theta) = r (k %2B 1) \cos \theta - r \cos \left( (k %2B 1) \theta \right) \,$

$y (\theta) = r (k %2B 1) \sin \theta - r \sin \left( (k %2B 1) \theta \right). \,$

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.

The epicycloid is a special kind of epitrochoid.

An epicycle with one cusp is a cardioid.

An epicycloid and its evolute are similar.[1]

1 Proof
2 See also
3 References
4 External links

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.