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An epicycle is a circle whose centre moves along another path, traditionally a larger circle called the deferent. Combining these motions can reproduce loops and changes of apparent speed in a planet's position on the celestial sphere. In Ptolemy's geocentric model, such constructions supplied quantitative planetary predictions.^[ref]

A simple planar example isThe second term is the epicycle relative to its moving centre. Suitable radii and angular rates can produce locally backward motion, illustrating apparent retrograde motion. The construction is a geometric representation; fitting an observed track does not by itself establish what force governs the actual orbit.

Ptolemy developed a mathematical system for predicting celestial motion in the Almagest by Ptolemy. The work combines a stationary central Earth with geometrical models and observational parameters for the apparent paths of the Sun, Moon and planets.

An epicycle is a circle whose center travels on another circle. Adding these motions can produce an apparent backward loop even though the component rotations continue steadily. This illustrates the predictive purpose of the construction: observed angular behavior is represented by organized component motions. A successful fit of sky directions does not by itself uniquely establish the physical arrangement of the bodies.^[ref]