You can use epicycles to draw the actual orbits of planets. That's what "epicycles" are, or were: a fairly accurate model of the observed orbits of planets relative to Earth built through a composition of circular motions.

One way to think about it is that you start by modeling the motion as a simple circular orbit. You then look at how the observed data differs from your circular model and add an epicycle whose parameters minimize the difference between your new model and your observations. Repeat this process until you get as close as you want to get.

The success of heliocentric elliptic-orbit model isn't that it's able to represent something impossible with epicycles, but that it's dramatically simpler. Kepler's three laws are:

1) The orbit of a planet is an ellipse with the Sun at one of the two foci.

2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.

3) The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

In contrast, the epicycle models had 40-80 cycles with parameters tuned to match the observed data.

I really like this video on how Fourier series work: https://www.youtube.com/watch?v=r6sGWTCMz2k

Kepler rule #2 never sunk in for me; the other two rules seemed fairly intuitive.

Kepler was Tycho Brahe's lab assistant, I think. Tycho was a curious chap. In addition to his brass nose, I believe he also had a "person of restricted stature" as a servant; that doesn't seem to be mentioned in his wikipedia article.