# Hypocycloid animation

Interactive demo of a smaller circle rolling inside of larger circle to create a hypocycloid

Imagine attaching a pen to a certain point on a small circle. If you were to then roll that small circle around the inside of a larger circle, the shape the pen would trace out would be a hypercycloid.

Mathematically speaking, the hypocycloid is defined by two parametric equations:

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

Where

• $$R$$ is the radius of the large circle
• $$r$$ is the radius of the small circle

The shape of the hypocycloid depends heavily on the ratio $$k = \frac{R}{r}$$. If $$k$$ is a whole number, then the hypocycloid will have $$k$$ sharp corners. For example, for $$k = 2$$ the shape will be a straight line (this system is known as the Tusi-couple, and was first investigated by Nasir al-Din al-Tusi in 1247). For $$k = 3$$, the hypocycloid will be a 3-pointed shape called a deltoid and for $$k = 4$$ it will be a 4-pointed shape called an astroid.

If $$k$$ is a fractional number, the number of corners is the numerator in simplest fractional representation of the ratio. So if the ratio is 1.67 recurring, the fraction will be $$\frac{5}{3}$$, so the hypocycloid has 5 corners.

This demo was inspired by http://en.wikipedia.org/wiki/File:Deltoid2.gif.