Epicycloid

Short description: Plane curve traced by a point on a circle rolled around another circle

The red curve is an epicycloid traced as the small circle (radius r = 1) rolls around the outside of the large circle (radius R = 3).

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 smaller circle has radius [math]\displaystyle{ r }[/math], and the larger circle has radius [math]\displaystyle{ R = kr }[/math], then the parametric equations for the curve can be given by either:

[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ z(\theta) = r \left( (k + 1)e^{ i\theta} - e^{i(k+1)\theta} \right) }[/math]

where

• the angle [math]\displaystyle{ \theta \in [0, 2\pi], }[/math]
• the smaller circle has radius [math]\displaystyle{ r }[/math], and
• the larger circle has radius [math]\displaystyle{ kr }[/math].

Area

(Assuming the initial point lies on the larger circle.) When [math]\displaystyle{ k }[/math] is a positive integer, the area of this epicycloid is

[math]\displaystyle{ A=(k+1)(k+2)\pi r^2. }[/math]

It means that the epicycloid is [math]\displaystyle{ \frac{(k+1)(k+2)}{k^2} }[/math] larger than the original stationary circle.

If [math]\displaystyle{ k }[/math] is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners).

If [math]\displaystyle{ k }[/math] is a rational number, say [math]\displaystyle{ k = p/q }[/math] expressed as irreducible fraction, then the curve has [math]\displaystyle{ p }[/math] cusps.

To close the curve and

complete the 1st repeating pattern :

θ = 0 to q rotations

α = 0 to p rotations

total rotations of outer rolling circle = p + q rotations

Count the animation rotations to see p and q

If [math]\displaystyle{ 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]\displaystyle{ R + 2r }[/math].

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

[math]\displaystyle{ R \leq \overline{OP} \leq R+2r }[/math]

where

• [math]\displaystyle{ R }[/math] = radius of large circle and
• [math]\displaystyle{ 2r }[/math] = diameter of small circle .

• Epicycloid examples
• 

  k = 1; a cardioid

• 

  k = 2; a nephroid

• 

  k = 3; a trefoiloid

• 

  k = 4; a quatrefoiloid

• 

  k = 2.1 = 21/10

• 

  k = 3.8 = 19/5

• 

  k = 5.5 = 11/2

• 

  k = 7.2 = 36/5

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

sketch for proof

We assume that the position of [math]\displaystyle{ p }[/math] is what we want to solve, [math]\displaystyle{ \alpha }[/math] is the angle from the tangential point to the moving point [math]\displaystyle{ p }[/math], and [math]\displaystyle{ \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

[math]\displaystyle{ \ell_R=\ell_r }[/math]

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

[math]\displaystyle{ \ell_R= \theta R }[/math]

and

[math]\displaystyle{ \ell_r= \alpha r }[/math].

From these two conditions, we get the identity

[math]\displaystyle{ \theta R=\alpha r }[/math].

By calculating, we get the relation between [math]\displaystyle{ \alpha }[/math] and [math]\displaystyle{ \theta }[/math], which is

[math]\displaystyle{ \alpha =\frac{R}{r} \theta }[/math].

From the figure, we see the position of the point [math]\displaystyle{ p }[/math] on the small circle clearly.

[math]\displaystyle{ 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]\displaystyle{ 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]

See also

Animated gif with turtle in MSWLogo (Cardioid)

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

References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 161,168–170,175. ISBN 978-0-486-60288-2. https://archive.org/details/catalogofspecial00lawr/page/161. 

External links

• Weisstein, Eric W.. "Epicycloid". http://mathworld.wolfram.com/Epicycloid.html. 
• "Epicycloid" by Michael Ford, The Wolfram Demonstrations Project, 2007
• O'Connor, John J.; Robertson, Edmund F., "Epicycloid", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Curves/Epicycloid.html .
• Animation of Epicycloids, Pericycloids and Hypocycloids
• Spirograph -- GeoFun
• Historical note on the application of the epicycloid to the form of Gear Teeth