The 3-dimensional extension of the 2-d ellipse. A generic ellipsoid should not be restricted to being a surface or solid of rotation, however. The points (x,y,z) which satisfy x2/a2+y2/b2+z2/c2=1 are the outer surface of a generic ellipsoid centered at the origin with parameters a, b and c.

The ellipse is defined to be a set of points (in the (x,y) coordinate system) each of whose distances from two fixed points sums to a constant. Similarly, the ellipsoid (being an ellipse in any planar section) is a set of points (x,y,z) each of whose distances from two points (no longer fixed; dependent upon direction!) sums to a constant. The two points in any particular direction are on the edge of an ellipse at the center of the ellipsoid. With this in mind, it can be said that "It takes two distinct points to specify a circle or a sphere; it takes three distinct points to specify an ellipse; it takes four distinct points to specify an ellipsoid."
A side note: since the circle is a specific case of the ellipse, Webster's definition could simply state "A solid, all plane sections of which are ellipses."