# Equation of ellipse formed by the projection of a tilted circle onto a plane

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 $$\beta$$.

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: $$x^2\cos^2\alpha+y^2(\sin^4\alpha+\cos^2\beta) -2xy\sin^2\alpha\cos\alpha=\omega^2\cos^2\alpha\cos^2\beta.$$