## Spirograph

A Hypotrochoid generated by a fixed point on a Circle rolling inside a fixed Circle. It has parametric equations,

 (1) (2)

where is the radius of the fixed circle, is the radius of the rotating circle, and is the offset of the edge of the rotating circle. The figure closes only if , , and are Rational. The equations can also be written
 (3) (4)

where the outer wheel has radius 1, the inner wheel a radius , the pen is placed units from the center, the beginning is at radians above the -axis, and
 (5) (6) (7) (8)

The following curves are for , with , 2, ..., 10, and .

Additional attractive designs such as the following can also be made by superposing individual spirographs.