# 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 rolling circle has radius $r$, and the fixed circle has radius $R=kr$, then the parametric equations for the curve can be given by either:
    ${\begin{aligned}&x(\theta )=(R+r)\cos \theta \ -r\cos \left({\frac {R+r}{r}}\theta \right)\\&y(\theta )=(R+r)\sin \theta \ -r\sin \left({\frac {R+r}{r}}\theta \right)\end{aligned}}$
or:
    ${\begin{aligned}&x(\theta )=r(k+1)\cos \theta -r\cos \left((k+1)\theta \right)\\&y(\theta )=r(k+1)\sin \theta -r\sin \left((k+1)\theta \right).\end{aligned}}$
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 rolling circle has radius $r$, and
* the fixed 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.
* To close the curve and
* complete the 1st repeating pattern :
* θ = 0 to q rotations
* α = 0 to p rotations
* total rotations of outer rolling circle = p + q rotations
Count the animation rotations to see p and q
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 .
* Epicycloid examples
* k = 1; a cardioid
* k = 2; a nephroid
* k = 3; a trefoiloid
* k = 4; a quatrefoiloid
* 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.

## 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.
    $x=\left(R+r\right)\cos \theta -r\cos \left(\theta +\alpha \right)=\left(R+r\right)\cos \theta -r\cos \left({\frac {R+r}{r}}\theta \right)$
    $y=\left(R+r\right)\sin \theta -r\sin \left(\theta +\alpha \right)=\left(R+r\right)\sin \theta -r\sin \left({\frac {R+r}{r}}\theta \right)$