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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."

El*lip"soid (?), n. [Ellipse + -oid: cf. F. ellipsoide.] Geom.

A solid, all plane sections of which are ellipses or circles. See Conoid, n., 2 (a).

The ellipsoid has three principal plane sections, a, b, and c, each at right angles to the other two, and each dividing the solid into two equal and symmetrical parts. The lines of meeting of these principal sections are the axes, or principal diameters of the ellipsoid. The point where the three planes meet is the center.

Ellipsoid of revolution, a spheroid; a solid figure generated by the revolution of an ellipse about one of its axes. It is called a prolate spheroid, or prolatum, when the ellipse is revolved about the major axis, and an oblate spheroid, or oblatum, when it is revolved about the minor axis.

El*lip"soid (?), El`lip*soi"dal (?), a.

Pertaining to, or shaped like, an ellipsoid; as, ellipsoid or ellipsoidal form.

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