Representing each planet's path as a combination of a cycle and an epicycle as seen here gives a quite accurate account of the paths of the planets through the heavens. It is based on the assumption of a stationary Earth, and a Sun orbiting the Earth. Press the button to see that the distance of Mars from the Sun remains constant throughout the motion, as it does in the correct modern model (ignoring for now the slight ellipticity of the orbit, which can be represented in the Ptolemaic scheme by small adjustments in the orbital speed.) The point is that the motion of Mars around its epicycle (the small circle) exactly mirrors the motion of the Sun in its cycle -- because both these motions are simply subtracting from the true motion the motion of the Earth in its orbit.

It's interesting to see how the path of Mars across the heavens is the same in this Ptolemaic scheme and in the (true) Copernican picture. To understand how this can be, check out this applet!

Code by Lawrence Hook.