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 $r$, and the larger circle has radius $R=kr$, then the parametric equations for the curve can be given by either:

$\begin{array}{rl}& x(\theta )=(R+r)\cos \theta -r\cos \left(\frac{R+r}{r}\theta \right)\\ & y(\theta )=(R+r)\sin \theta -r\sin \left(\frac{R+r}{r}\theta \right)\end{array}$

or:

$\begin{array}{rl}& x(\theta )=r(k+1)\cos \theta -r\cos ((k+1)\theta )\\ & y(\theta )=r(k+1)\sin \theta -r\sin ((k+1)\theta ).\end{array}$

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

$z(\theta )=r((k+1)e^{i\theta }-e^{i(k+1)\theta })$

where

• the angle $\theta \in [0,2\pi ],$
• the smaller circle has radius $r$, and
• the larger circle has radius $kr$.

Area

(Assuming the initial point lies on the larger circle.) When $k$ is a positive integer, the area of this epicycloid is

$A=(k+1)(k+2)\pi r^2.$

It means that the epicycloid is $\frac{(k+1)(k+2)}{k^2}$ larger than the original stationary circle.

If $k$ is a positive integer, then the curve is closed, and has k 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.

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

$R\le \overline{OP}\le R+2r$

where

• $R$ = radius of large circle and
• $2r$ = 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

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sketch for 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

$\alpha =\frac{R}{r}\theta$.

From the figure, we see the position of the point $p$ on the small circle clearly.

$x=(R+r)\cos \theta -r\cos (\theta +\alpha )=(R+r)\cos \theta -r\cos \left(\frac{R+r}{r}\theta \right)$

$y=(R+r)\sin \theta -r\sin (\theta +\alpha )=(R+r)\sin \theta -r\sin \left(\frac{R+r}{r}\theta \right)$

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