# 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?

• So how exactly did you try to do the projection and how does that not work? Show us what you did Jul 7, 2020 at 15:13
• @planetmaker I drew the ellipses for each orbit orienting the major axes using the Argument of Periapsis, and then redrew them using the Longitude of the Ascending Node but neither of the results matched the diagram from the paper I referenced, so I am now assuming that I need to combine both the angles somehow, but I'm not sure how Jul 7, 2020 at 16:04
• @jmh is right about working with x, y, and z coordinates, but I have also devised a script to draw the ellipse from any angle. I’m willing to share if you still need it. Nov 30, 2020 at 0:25
• That sounds useful @PierrePaquette if you could share it then that would be great Nov 30, 2020 at 10:03

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.