Representing the planet's path as a combination of a cycle and an epicycle as seen here gives a reasonably 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. Notice that in this model, the distance of Mars from the Sun remains constant throughout the motion, as it does in the correct modern model (if we ignore the slight departure from a circle of Mars' orbit). For further discussion, check out my lecture here.
Code by Lawrence Hook.