Ellipsoidal coordinates
Encyclopedia
Ellipsoidal coordinates are a three-dimensional orthogonal
coordinate system
that generalizes the two-dimensional elliptic coordinate system
. Unlike most three-dimensional orthogonal
coordinate system
s that feature quadratic
coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.
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.
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
Orthogonal coordinates
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
that generalizes the two-dimensional elliptic coordinate systemElliptic coordinates
In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in whichthe coordinate lines are confocal ellipses and hyperbolae...
. Unlike most three-dimensional orthogonal
Orthogonal coordinates
In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
coordinate system
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
s that feature quadratic
Quadratic
In mathematics, the term quadratic describes something that pertains to squares, to the operation of squaring, to terms of the second degree, or equations or formulas that involve such terms...
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 equationswhere the following limits apply to the coordinates
Consequently, surfaces of constant
are ellipsoidswhereas surfaces of constant
are hyperboloids of one sheetbecause the last term in the lhs is negative, and surfaces of constant
are hyperboloids of two sheetsbecause the last two terms in the lhs are negative.
Scale factors and differential operators
For brevity in the equations below, we introduce a functionwhere
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
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Other differential operators such as
and
can be expressed in the coordinates 
by substituting
the scale factors into the general formulae
found in orthogonal coordinatesOrthogonal coordinatesIn mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles . A coordinate surface for a particular coordinate qk is the curve, surface, or hypersurface on which qk is a constant...
.
External links
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