Hypocycloid animation

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.