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:
:$\begin & x (\theta) = (R + r) \cos \theta \ - r \cos \left( \frac \theta \right) \\ & y (\theta) = (R + r) \sin \theta \ - r \sin \left( \frac \theta \right) \end$
or:
:$\begin & x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \\ & y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right). \end$

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

:$z(\theta) = r \left( (k + 1)e^ - e^ \right)$

where
* the angle \theta \in , 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 $\frac$ 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.

Count the animation rotations to see and

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

:$R \leq \overline \leq R+2r$
where
*$R$ = radius of large circle and
*$2r$ = diameter of small circle .

File:Epicycloid-1.svg, ; a ''cardioid''
File:Epicycloid-2.svg, ; a ''nephroid''
File:Epicycloid-3.svg, ; a ''trefoiloid''
File:Epicycloid-4.svg, ; a ''quatrefoiloid''
File:Epicycloid-2-1.svg,
File:Epicycloid-3-8.svg,
File:Epicycloid-5-5.svg,
File:Epicycloid-7-2.svg,

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
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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
:$\alpha =\frac \theta$.

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

See also

* List of periodic functions
* Cycloid
* Cyclogon
* Deferent and epicycle
* Epicyclic gearing
* Epitrochoid
* Hypocycloid
* Hypotrochoid
* Multibrot set
* Roulette (curve)
* Spirograph

References

*

External links

*

Epicycloid
by Michael Ford, The Wolfram Demonstrations Project, 2007
*{{MacTutor, class=Curves, id=Epicycloid, title=Epicycloid
Animation of Epicycloids, Pericycloids and HypocycloidsSpirograph -- GeoFunHistorical note on the application of the epicycloid to the form of Gear Teeth
Algebraic curves
Roulettes (curve)