Suppose we have a slice of an ellipsoid parallel to the major axis (but not on the major axis) so that we get a concave ellipsoidal mirror. I have knowledge of just the slice and nothing about the ellipsoid itself with measurements of the slice's 'minor/major' axis (say $\mathit{a'}$ and $\mathit{b'}$ ) and the depth of the slice (from the center of the ellipsoidal slice to the plane where it is sliced) say $\mathit{z'}$ .
Is it possible to reconstruct the original ellipsoid from just this information?
My intuition is telling me that it should be possible to reconstruct the ellipsoid so that I can find the foci (in one plane) of the ellipsoid since there will be only one solution to fit the 3 measurements made on the slice.