I am looking into a circle rotated with respect to the line of sight by angle $\alpha$ around the vertical axis and then and $\beta$ around the horizontal axis, so instead of a circle, I see an ellipse. I know the ratio $a$ between major and minor axis of the ellipse, as well as the angle $\theta$ that the major axis of the ellipse makes with the horizon, but the size of the circle and distance to it are not know with a useful precision.
If the circle is rotated around the vertical axis only ($\beta=0$), $\theta=90$ (vertical) and probably I can use the formula of the cross section of the cylinder to find $\alpha$ as described here; $\beta=0$. The similar solution works when rotating around the horizontal axis only.
By moving a dish plate in front of my eyes, I see that when $\beta=45, \alpha=45$ the major axis of the ellipse goes at angle, probably $\theta=+45$ or $\theta=-45$, depending on how exactly do I rotate, while aspect ratio $a$ shrinks. Is it possible to find the two rotation angles $\alpha$ and $\beta$ from the visible shape of the ellipse?