Epicycloid

In geometry, an epicycloid or 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.

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).

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:

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

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

or:

$x(\theta )=r(k+1)\cos \theta -r\cos \left((k+1)\theta \right)\,$

$y(\theta )=r(k+1)\sin \theta -r\sin \left((k+1)\theta \right).\,$

(Assuming the initial point lies on the larger circle.)

If k is a positive integer, then the curve is closed, and has k cusps (i.e., sharp corners, where the curve is not differentiable).

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 OP from (x=0,y=0) origin to (the point $p$ on the small circle) varies up and down as

R <= OP <= (R + 2r)

R = radius of large circle and

2r = diameter of small circle

• Epicycloid examples
• 
  k = 1          a cardioid
• 
  k = 2          a nephroid
• 
  k = 3
• 
  k = 4
• 
  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 $p$ is what we want to solve, $\alpha$ is the radian from the tangential point to the moving point $p$, and $\theta$ is the radian 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 radian (which is the rate arc over radius), then we have that

$\ell _{R}=\theta R,\ell _{r}=\alpha r$

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.

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

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

See also

• 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.

External links

• Weisstein, Eric W. "Epicycloid". MathWorld.
• "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
• Animation of Epicycloids, Pericycloids and Hypocycloids
• Spirograph -- GeoFun
• Historical note on the application of the epicycloid to the form of Gear Teeth