How to calculate shape of orbit when projected onto the plane of the ecliptic I'm trying to generate an 'overhead' view of the orbits of the moons of Neptune, similar to Figure 4 in this paper, using values given here. I initially tried using both the Argument of Periapsis, and the Longitude of the Ascending Node separately but neither seems to give the correct results. After a bit more reading I now think I will need to combine both these values to derive the orientation of the major axis of the ellipse, but I can't find the formula anywhere. Am I correct? Could someone show me the correct calculation?
You could convert the orbital elements into x,y,z and $ \dot{x}, \dot{y}, \dot{z}$ and then calculate a whole orbit. A source for converting orbital elements into cartesian coordinates is here.