Epicycloid

thumb|500px|The red curve is an epicycloid traced as the small circle (radius rolls around the outside of the large circle (radius .

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 <math>r</math>, and the fixed circle has radius <math>R = kr</math>, then the parametric equations for the curve can be given by either:

:<math>\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}</math>

or:

:<math>\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}</math>

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

:<math>z(\theta) = r \left( (k + 1)e^{ i\theta} - e^{i(k+1)\theta} \right) </math>

where

the angle <math>\theta \in [0, 2\pi],</math>
the rolling circle has radius <math>r</math>, and
the fixed circle has radius <math>kr</math>.

Area and arc length

Assuming the initial point lies on the larger circle, when <math>k</math> is a positive integer, the area <math>A</math> and arc length <math>s</math> of this epicycloid are

:<math>A=(k+1)(k+2)\pi r^2,</math>

:<math>s=8(k+1)r.</math>

It means that the epicycloid is <math>\frac{(k+1)(k+2)}{k^2}</math> larger in area than the original stationary circle.

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

If <math>k</math> is a rational number, say <math>k = p/q</math> expressed as irreducible fraction, then the curve has <math>p</math> cusps.

{| class="wikitable"

|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 <math>k</math> 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 <math>R + 2r</math>.

The distance <math>\overline{OP}</math> from the origin to the point <math>p</math> on the small circle varies up and down as

:<math>R \leq \overline{OP} \leq R+2r </math>

where

<math>R</math> = radius of large circle and
<math>2r</math> = 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|

</gallery>

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

thumb|upright=2.0|sketch for proof

Assuming that the position of <math>p</math> is what has to be solved, <math>\alpha</math> is the angle from the tangential point to the moving point <math>p</math>, and <math>\theta</math> is the angle from the starting point to the tangential point.

Since there is no sliding between the two cycles, then

:<math>\ell_R=\ell_r</math>

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

:<math>\ell_R= \theta R</math>

and

:<math>\ell_r= \alpha r</math>.

From these two conditions, the following identity is obtained

:<math>\theta R=\alpha r</math>.

By calculating, the relation between <math>\alpha</math> and <math>\theta</math> is obtained, which is

:<math>\alpha =\frac{R}{r} \theta</math>.

From the figure, the position of the point <math>p</math> on the small circle is clearly visible.

:<math> 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)</math>

:<math>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)</math>

Equations

Area and arc length

Proof

See also

References