Solar System Simulations
When I was in high school, I used to program two dimensional solar system simulations for fun (as you can surely tell, I was extremely popular and desired by all the ladies far and wide). I would put a "sun" in the middle of the screen and I'd spin "planets" around in simulated orbits.
There are essentially two ways you can go about programming solar system simulations. Technique 1, which for lack of a better term I will call the explicit technique, means deciding (or realizing) at the outset that the orbits of your planets will be elliptical, and then writing your program based on that. The explicit technique works because there are certain exploitable patterns in elliptical planetary orbits that can be harnessed to great advantage. Planets move faster when they are closer to the sun, for example, and slower when they are further away, in mathematically determined ways. One can derive formulae for the position of the planet on the orbit versus time.