Ellipsoidal coordinates

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 not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

Basic formulae

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.

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.

See also



    Unusual convention

    External links

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