Epicycloid

In geometry, an epicycloid 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.

Equations

If the smaller circle has radius , and the larger circle has radius , 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:
:$\begin & x (\theta) = r (k + 1) \cos \theta - r \cos \left( (k + 1) \theta \right) \\ & y (\theta) = r (k + 1) \sin \theta - r \sin \left( (k + 1) \theta \right) \end$

in a more concise and complex form

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

where
* angle is in turns: \theta \in , 2\pi
* smaller circle has radius
* the larger circle has radius

Area

(Assuming the initial point lies on the larger circle.) When is a positive integer, the area of this epicycloid is
:$A=(k+1)(k+2)\pi r^2.$

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

If is a rational number, say expressed as irreducible fraction, then the curve has cusps.

Count the animation rotations to see and

If 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 .

The distance from origin to (the point on the small circle) varies up and down as

:$R \leq \overline \leq R+2r$
where
* = radius of large circle and
* = 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
/ref>

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)