Three-dimensional coordinate system
Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system formula_1 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.
Basic formulae.
The Cartesian coordinates formula_2 can be produced from the ellipsoidal coordinates 
formula_3 by the equations
formula_4
formula_5
formula_6
where the following limits apply to the coordinates
formula_7
Consequently, surfaces of constant formula_8 are ellipsoids
formula_9
whereas surfaces of constant formula_10 are hyperboloids of one sheet
formula_11
because the last term in the lhs is negative, and surfaces of constant formula_12 are hyperboloids of two sheets
formula_13
because the last two terms in the lhs are negative.
The orthogonal system of quadrics used for the ellipsoidal coordinates are confocal quadrics.
Scale factors and differential operators.
For brevity in the equations below, we introduce a function
formula_14
where formula_15 can represent any of the three variables formula_16. 
Using this function, the scale factors can be written
formula_17
formula_18
formula_19
Hence, the infinitesimal volume element equals
formula_20
and the Laplacian is defined by
formula_21
Other differential operators such as formula_22 and formula_23 can be expressed in the coordinates formula_1 by substituting the scale factors into the general formulae found in orthogonal coordinates.
Angular parametrization.
An alternative parametrization exists that closely follows the angular parametrization of spherical coordinates:
formula_25
formula_26
formula_27
Here, formula_28 parametrizes the concentric ellipsoids around the origin and formula_29 and formula_30 are the usual polar and azimuthal angles of spherical coordinates, respectively. The corresponding volume element is
formula_31
References.
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