The "major axis" of the elliptical orbit is $ \ 2a \ $ , which is the sum of the perihelion distance $ \ r_p \ $ and the aphelion distance $ \ r_a \ $ . Since the "central body" (the Sun, in this case) is at one focus of the ellipse, we have $ \ r_p \ = \ a - c \ $ and $ \ r_a \ = \ a + c \ $ .
Since $ \ c \ = \ ea \ , $ we can also write $ \ r_p \ = \ a \ (1 - e) \ $ and $ \ r_a \ = \ a \ (1 + e ) \ , $ with $ \ 0 \ \le \ e \ < 1 \ $ for elliptical orbits.
For tracing the entire ellipse around the focus where the Sun lies, you can use the (focal) polar equation for the ellipse,
$$ r(\theta) \ = \ \frac{a \ (1-e^2)}{1 \ + \ e \cos \theta} \ \ , $$
with the angle $ \ \theta \ $ being measured from the direction of perihelion.
If you are making a two-dimensional map and want it to be more accurate, you will also want the direction of perihelion for each orbit, as measured from a reference direction (0ยบ is conventionally taken as the direction against the stars of the vernal equinox). This will tell you how each orbital major axis is oriented.
If you are making a three-dimensional model, you will also need the orbital inclinations (the angle of tilt relative to either the "ecliptic" [the plane of Earth's orbit] or to the mean plane of the Solar System) and the directions of the ascending nodes (where each orbit crosses from below to above the reference plane).
