Spirograph

Topic:
Circle
There's an old art toy called a spirograph that made beautiful patterns by rotating circular gears inside and outside circles and tracing their path. That's what this sketch does. Mathematicians call these curves cycloids. Inside a circle is a hypocycloid and outside is an epicycloid. They are specific types of a more general curve family know as trochoids. That's when you attach the tracing point to a radial arm on the rolling circle. Mathematically, can you determine what the controls do? How they result in the picture they make? How many circuits (times around the circle) it will take to get back to the beginning? For the art you make, what is interesting about it mathematically? Remember that GeoGebra can export pictures; if you uncheck inside and outside, it will also hide the drawing tools.
More GeoGebra at mathhombre.blogspot.com.