Epicycloid

The red curve is an epicycloid traced as the small circle (radius Template:Math rolls around the outside of the large circle (radius Template:Math.

In geometry, an epicycloid (also called hypercycloid)<ref>[1]</ref> 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.

EquationsEdit

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<ref>Epicycloids and Blaschke products by Chunlei Cao, Alastair Fletcher, Zhuan Ye</ref>

<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 LengthEdit

(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

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 Template:Mvar 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.

To close the curve and
complete the 1st repeating pattern :
Template:Math to Template:Mvar rotations
Template:Math to Template:Mvar rotations
total rotations of outer rolling circle = Template:Math rotations

Count the animation rotations to see Template:Mvar and Template:Mvar

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 .

Epicycloid examples
Epicycloid-1.svg

Template:Math; a cardioid
Epicycloid-2.svg

Template:Math; a nephroid
Epicycloid-3.svg

Template:Math; a trefoiloid
Epicycloid-4.svg

Template:Math; a quatrefoiloid
Epicycloid-2-1.svg

Template:Math
Epicycloid-3-8.svg

Template:Math
Epicycloid-5-5.svg

Template:Math
Epicycloid-7-2.svg

Template:Math

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.<ref>Epicycloid Evolute - from Wolfram MathWorld</ref>

ProofEdit

File:Epizykloide herleitung.svg

sketch for proof

We assume that the position of <math>p</math> is what we want to solve, <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 we have that

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

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

and

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

From these two conditions, we get the identity

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

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

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

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

<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>

ReferencesEdit

Template:Cite book

External linksEdit

{{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Epicycloid%7CEpicycloid.html}} |title = Epicycloid |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}