A spirograph is a toy for drawing pictures. Basically, it is a collection of gears. Put two gears down on a piece of paper, push a pen through a hole in one of them, and trace a curve as the one gear runs around the other.
In this applet you can specify the radius of a large gear (Fixed radius) to be thought of as a thin annulus. It has teeth on both the inside and the outside. This large gear is held fixed. You can also specify the radius of the gear that moves (Moving radius) and how far from the center of the smaller gear the pen is placed (Offset). The Initial angle (in radians) and the Inside/Outside radiobuttons give the starting position of the moving circle. Select a color and how long a curve you want to draw (Interval) and click the Draw button.
Marion and Larry Aplin sent me a spirograph from Geneva when I was a little kid. I had no idea where Geneva. But I turned into a demon for spirograph curves anyway, spending hours working out rules for how many loops a curve had before closing in terms of the numbers of teeth on the gears.
If you want, you can have a copy of the source for this program.