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."