So I'm assuming that the x-y plane is represented by the normal vector (0,0,1), and I have a circle with radius $\omega$, given by ($\omega$ cos[$\phi$], $\omega$ sin[$\phi$], 0).
I apply rotations about the x and y axis of $\alpha$ and $\beta$ respectively to the circle and project it onto the x-y plane. Which gives me the parametric form of an ellipse:
($\omega$ (cos[$\beta$] cos[$\phi$] +sin$^2$[$\alpha$] sin[$\phi$]), $\omega$ cos[$\alpha$] sin[$\phi$], 0)
All is well so far, I plotted this with a couple of tilt angles and things seem to behave as expected. However, I would like to fit this to 5 data pairs, for which I need a non-parametric version. Is anyone able to massage this into the equation of a rotated ellipse? I need the ellipse to be described by the tilt angles $\alpha$ and $\beta$.