**What shape does the moon trace as the Earth orbits the Sun?**

The idea is, the Moon is going in (approximately) a circle around the Earth, while the Earth is going in (approximately) a circle around the Sun. This is a surprisingly simple situation to model using a parametric equation:

The general form for this type of curve is

We can relate this to the earlier equations with

The site I linked above states that there will be loops if b > a/c, so let's plug in some values (from Wikipedia):Nowhere close, a result borne out by the plot:

If you look carefully, you can see that's not quite circular, but it's awfully close. Thanks for a great question, George!

[Edit: If you're curious about the b > a/c condition, you can think of it this way – There will be a loop anytime the Moon is moving backwards faster than the Earth is moving forwards. The speed of each is given by 2πR/T, and if you plug in the R and T relations for a, b, and c, you'll see that's exactly what b > a/c means.]