ELLIPSOID, a quadric surface whose sections are ellipses. Analytically, it has for its equation x2/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 proportional to 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. (See Mechanics.)