Epicycloid

In geometry, an epicycloid is a plane curve produced by tracing the path of a chosen point on the circumference 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 (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.

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, two cusps is a nephroid.

An epicycloid and its evolute are similar.

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

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