The ellipsoid is a linear deformation of the sphere.
The rotationally symmetric ellipsoid has an interesting focussing property, called “wisper room”. If some pub has a ceiling of this shape and a table is placed at one focal point then a spy can hear from a safe distance the conversations at the table.
The focal rays again.
The coordinate planes are symmetry planes of the ellipsoid and therefore they intersect it in closed geodesics. These are ellipses. All other closed geodesics are much more complicated. Here is one in anaglyph.
Another closed geodesic on a raytraced ellipsoid. Note that the pieces behind the ellipsoid are in darker color.
Conic Section Surfaces
Hyperboloid of One Sheet
Hyperboloid of Two Sheet
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3DXM Surface Gallery