ELLIPSOID  , a

quadric surface whose sections are ellipses . Analytically, it has for its equation xz/a2+y2/b2+z2/c2 =1, a, b, c being its axes; the name is also given to the solid contained by this surface (see GEOMETRY: Analytical) . The solids and surfaces of revolution of the ellipse are sometimes termed ellipsoids, but it is advisable to use the name spheroid (q.v.) . The ellipsoid appears in the mathematical investigation of physical properties of media in which the particular property varies in three directions within the media; such properties are the elasticity, giving rise to the strain ellipsoid, thermal expansion, ellipsoid of expansion, thermal conduction, refractive index (see CRYSTALLOGRAPHY), &c . In mechanics, the ellipsoid of gyration or inertia is such that the perpendicular from the centre to a tangent plane is equal to the radius of gyration of the given body about the perpendicular as axis; the " momental ellipsoid," also termed the " inverse ellipsoid of inertia " or Poinsot's ellipsoid, has the perpendicular inversely proportionalto the radius of gyration; the " equimomental ellipsoid " is such that its moments of inertia about all axes are the same as those of a given body .