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.

## 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:

or:

(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).

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* + 2*r*.

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

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

R = radius of large circle and

2r = diameter of small circle

*k*= 1 a*cardioid**k*= 2 a*nephroid**k*= 3 - resembles a*trefoil**k*= 4 - resembles a*quatrefoil*

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.^{[1]}

## Proof

We assume that the position of is what we want to solve, is the radian from the tangential point to the moving point , and is the radian 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 radian (which is the rate arc over radius), then we have that

From these two conditions, we get the identity

By calculating, we get the relation between and , which is

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

## 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