SpiroGraph

The path of a point fixed relative to a circle that rolls along a straight line is called a trochoid. The easiest way to visualize this phenomenon is to think of the path of a reflector on a bicycle as someone is riding on a level street. The reflector rotates around the hub of the wheel, but yet the hub of the wheel is moving relative to the ground. Here is an applet that demonstrates this (without the bicycle)

Now think about what happens if the circle is instead rotating around another circle. In other words, the cyclist is now pedaling his/her way around the equator on a bike with very large wheels. Mathematicians call this path an epicycloid. The rest of the world calls them SpiroGraphs!! The parametric equations for these curves are given by:

x(t)=(R+r)cos(t) + p*cos((R+r)t/r)
y(t)=(R+r)sin(t) + p*sin((R+r)t/r)