Wiki: Epicycloid - upcScavenger

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 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{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 Epicycloids and Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye

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

where

the angle $\theta \in 0,,$
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 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 :

to rotations

total rotations of outer rolling circle = rotations

Count the animation rotations to see and

If $k$ is an irrational number, then the curve never closes, and forms a dense set 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. Epicycloid Evolute - from Wolfram MathWorld

Proof

Assuming that the position of

p

is what has to be solved,

\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

\ell_R=\ell_r

By the definition of angle (which is the rate arc over radius), then

\ell_R= \theta R

and

\ell_r= \alpha r

.

From these two conditions, the following identity is obtained

\theta R=\alpha r

.

By calculating, the relation between

\alpha

and

\theta

is obtained, which is

\alpha =\frac{R}{r} \theta

.

From the figure, the position of the point $p$ on the small circle is clearly visible.

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)

Epicycloid

( Algebraic Curves )

Account

Navigation

Statistics

Epicycloid ( Algebraic Curves )

Account

Navigation

Statistics

Epicycloid

( Algebraic Curves )