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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 $\beta$.

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You can rewrite your parametric equations as follows: $$ x-y{\sin^2\alpha\over\cos\alpha}=\omega\cos\beta\cos\phi, \quad y=\omega \cos\alpha \sin\phi. $$ Dividing these, respectively, by $\cos\beta$ and $\cos\alpha$, then squaring and adding together, we finally get: $$ x^2\cos^2\alpha+y^2(\sin^4\alpha+\cos^2\beta) -2xy\sin^2\alpha\cos\alpha=\omega^2\cos^2\alpha\cos^2\beta. $$

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