that generalizes the two-dimensional elliptic coordinate system.
Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is based on confocal quadrics.
by the equations where the following limits apply to the coordinates Consequently, surfaces of constant
are hyperboloids of one sheet because the last term in the lhs is negative, and surfaces of constant
are hyperboloids of two sheets because the last two terms in the lhs are negative.
For brevity in the equations below, we introduce a function where
Using this function, the scale factors can be written Hence, the infinitesimal volume element equals and the Laplacian is defined by Other differential operators such as
by substituting the scale factors into the general formulae found in orthogonal coordinates.
parametrizes the concentric ellipsoids around the origin and
are the usual polar and azimuthal angles of spherical coordinates, respectively.