What is this "Tusi Couple"?
The ancients widely accepted Plato's assertion that the only perfect motion was circular motion at a constant speed, so the rather complicated motion of the planets through the night sky must be some combination of such motions. In particular, Ptolemy's model of cycles and epicycles was very successful at predicting planetary motion over hundreds of years. Nevertheless, it wasn't perfect: careful observation revealed a small component of the motion that was up and down in a straight line. (We now understand this as due to the planetary orbits being in slightly different planes, so as a planet goes around, it goes first a little above, then below, the Earth's orbital plane.)
In the 1260's, Al Tusi, working in a magnificent observatory built for him by the Mongols in northern Iran, created a circular model, a combination of circular motions, that gave the desired up-down movement. This is the Tusi couple illustrated by this applet, and recorded in this Vatican manuscript.
My lecture on later Islamic science can be found here.
Geometrically, the Tusi Couple is a special case of a hypocycloid.
Code by Casey Bowler