Epicycloid

In geometry, an epicycloid (also called hypercycloid)^[1] 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{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^[2]

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.

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.^[3]

Proof

sketch for 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.