Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system 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.
The Cartesian coordinates can be produced from the ellipsoidal coordinates by the equations
where the following limits apply to the coordinates
Consequently, surfaces of constant are ellipsoids
whereas 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.
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
where can represent any of the three variables . 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 and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates.
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