Given that I know the coordinates of one focus (the central body) and two points on the orbit (two vectors) how do I calculate the argument of periapsis? And Longitude of the ascending node as well? All of this in 3D space. The reference plane is the plane z = 0 and the reference direction is the x-axis.
What I want to accomplish is to draw the ellipse in 3D and for that, I need the Keplerian elements. I have followed the direction on Lambert's Problem to calculate the semi-major axis and eccentricity. From my own geometry knowledge, I have calculated the inclination. The remaining ones are the Longitude of the ascending nodeand the Argument of periapsis both of which I have tried to find a way to calculate but to no success. Both of them seem to require a vector pointing to the ascending node and one of them requires a vector pointing to the periapsis. It's here where I'm getting stuck on how to calculate those two vectors given just two points on the ellipse (and not a position and velocity).
Yes, I do know the travel time and the mass of the central body. I know basically everything except the argument of periapsis and the longitude of the ascending node.
Clarification: Velocity vectors at the two points are also unknown.