
The astroid is a famous curve that arises in an interesting way. Suppose that a ball of radius a/2 is rolling inside a ball of radius a. The astroid is the curve traced out by a point on the inner circle. You can find a java applet demonstrating this at http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/derivative/implicit.html. Here is that applet. You can see the curve being swept out by either dragging the red ball around or by starting the animation using the "Start" button.