Deferent and epicycle (wkp) & via stub for "equant":

Ptolemaic System - at Rice University's Galileo Project goodlooking long article w illustrations
Ptolemy used three basic constructions, the eccentric, the epicycle, and the equant.
An eccentric construction is one in which the Earth is placed outside the center of the geometrical construction. Here, the Earth, E, is displaced slightly from the center, C, of the path of the planet. Although this construction violated the rule that the Earth was the center of the cosmos and all planetary motions, the displacement was minimal and was considered a slight bending of the rule rather than a violation. The eccentric in the figure below is fixed; it could also be made movable. In this case the center of the large circle was a point that rotated around the Earth in a small circle centered on the Earth. In some constructions this little circle was not centered in the Earth.
The second construction, the epicycle, is geometrically equivalent to the simple movable eccentric. In this case, the planet moved on a little circle the center of which rotated on the circumference of the large circle the deferent centered on the Earth. When the directions and speeds of rotation of the epicycle and large circle were chosen appropriately, the planet, as seen from the Earth, would stop, reverse its course, and then move forward again. right good. Thus the annual retrograde motion of the planets (caused, in heliocentric terms good heliocentric terms rather than "really", by the addition of the Earth's annual motion to the motion of the planet) could roughly be accounted for.
But these two constructions did not quite bring the resulting planetary motions within close agreement with the observed motions. Ptolemy therefore added yet a third construction, the equant. In this case, the center of construction (the geometrical center) of the large circle was separated from the center of motion of a point on its circumference weird this doesnt come immediately back to my understanding, as shown below, where C is the geometrical center of the large circle (usually called in these constructions the excentric circle) but the motion of the center of the epicycle, P (middle figure), is uniform about Q, the equant point (righthand side figure). ok right so effectively it serves to explains change in speed (not regular - perfect- in motion relative to the geometrical ctr of the circle, so pick the point relative to wh it can be said to be perfect regular circular motion. uniform about Q, the equant point.