# Rolling Hypocycloids and Epicycloids

 Press a to make more shapes, or click on the icon. Press s to make fewer shapes, or click on the icon. Press w to change colours, or click on the icon. Move the mouse (or touch drag) to roll the shapes.

## Explanation

Hypocycloids and epicycloids are curves traced out by a point on a circle rolling on the inside or outside of another circle. These curves are special cases of hypotrochoids and epitrochoids (the cases when $$d=r$$ using the notation from the side Make a Spirograph).

Let $$r$$ be the radius of the rolling circle and $$R$$ the radius of the fixed circle, then all curves having the same ratio $$R:r$$ have the same shape. When the ratio $$R:r$$ is an integer, as in the GeoGebra-example below, the curve traced out as the smaller circle rolls one revolution around the large circles is a closed curve. Furthermore, if the ratio is an integer $$n$$ then the curves have $$n$$ edges called cusps.

By letting the rolling circles have a smaller radius, thus smoothening the cusps, distorted versions of the curves are created

In the GeoGebra-example above, the equation for the hypocycloid is

$\begin{cases} x(t) = \dfrac{(n - 1) \cos(t) + \cos((n - 1) t)}{n} \\ y(t) = \dfrac{(n - 1) \sin(t) - \sin((n - 1) t)}{n} \end{cases}$

and for the epicycloid it is

$\begin{cases} x(t) = \dfrac{(n + 1) \cos(t) - \cos((n + 1) t)}{n} \\ y(t) = \dfrac{(n + 1) \sin(t) - \sin((n + 1) t)}{n}. \end{cases}$

### Rolling on the inside

A hypocycloid (or epicycloid) with $$n$$ cusps can move inside a hypocycloid (or epicycloid) with $$n+1$$ cusps in such a way that the cusps of one of the curves always touches the other curve.

By repeating this pattern, examples such as the Javascript-examples above can be made.

### Rolling on the outside

The cusps of a $$(n+1)$$-hypocycloid, rolling on the outside of a $$n$$-hypocycloid, traces out a $$n$$-epicycloid.

Explanation:
If you roll a $$n$$-epicycloid on the inside of a $$(n+1)$$-epicycloid, the cusps of the outer epicycloid lie on the inner epicycloid. So if you instead roll epicycloids on the outside, the cusps of the $$(n+1)$$-epicycloid (red below) will glide along the inner $$n$$-epicycloid. Since the cusps of $$(n+1)$$-epicycloids are the same points as the cusps of $$(n+1)$$-hypocycloids, we get that the cusps of a hypocycloid outlines an epicycloid when rolling on the outside.

## Animated gifs

Distorted epicycloids and hypocycloids on tumblr.

Rolling epicycloids (black and white) on tumblr.

Rolling hypocycloids (black and white) on tumblr.

## References

Ideas from the discussion by John Baez and others at: Azimuth - Rolling Hypocycloids.

GeoGebra-examples at geogebra.org: