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!