Epicycloid

In geometry, an epicycloid (also called hypercycloid) 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.

An epicycloid with a minor radius (R2) of 0 is a circle. This is a degenerate form.

Equations
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:
 * $$\begin{align}

& 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) \end{align}$$ or:
 * $$\begin{align}

& 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). \end{align}$$

This can be written in a more concise form using complex numbers as


 * $$z(\theta) = r \left( (k + 1)e^{ i\theta} - e^{i(k+1)\theta} \right) $$

where
 * the angle $$\theta \in [0, 2\pi],$$
 * the smaller circle has radius $$r$$, and
 * the larger circle has radius $$kr$$.

Area and Arc Length
(Assuming the initial point lies on the larger circle.) When $$k$$ is a positive integer, the area $$A$$ and arc length $$s$$ of this epicycloid are
 * $$A=(k+1)(k+2)\pi r^2,$$
 * $$s=8(k+1)r.$$

It means that the epicycloid is $$\frac{(k+1)(k+2)}{k^2}$$ larger in area than the original stationary circle.

If $$k$$ is a positive integer, then the curve is closed, and has $k$ cusps (i.e., sharp corners).

If $$k$$ is a rational number, say $$k = p/q$$ expressed as irreducible fraction, then the curve has $$p$$ cusps. Count the animation rotations to see $q$ and $p$

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 distance $$\overline{OP}$$ from the origin to the point $$p$$ on the small circle varies up and down as


 * $$R \leq \overline{OP} \leq R+2r $$

where
 * $$R$$ = radius of large circle and
 * $$2r$$ = diameter of small circle.

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.

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

Since there is no sliding between the two cycles, then we have that
 * $$\ell_R=\ell_r$$

By the definition of angle (which is the rate arc over radius), then we have that
 * $$\ell_R= \theta R$$

and
 * $$\ell_r= \alpha r$$.

From these two conditions, we get the identity
 * $$\theta R=\alpha r$$.

By calculating, we get the relation between $$\alpha$$ and $$\theta$$, which is
 * $$\alpha =\frac{R}{r} \theta$$.

From the figure, we see the position of the point $$p$$ on the small circle clearly.
 * $$ x=\left( R+r \right)\cos \theta -r\cos\left( \theta+\alpha \right) =\left( R+r \right)\cos \theta -r\cos\left( \frac{R+r}{r}\theta \right)$$
 * $$y=\left( R+r \right)\sin \theta -r\sin\left( \theta+\alpha \right) =\left( R+r \right)\sin \theta -r\sin\left( \frac{R+r}{r}\theta \right)$$