Elliptic coordinate system

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In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system.

Elliptic coordinate system

Basic definition

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The most common definition of elliptic coordinates   is

 

where   is a nonnegative real number and  

On the complex plane, an equivalent relationship is

 

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

 

shows that curves of constant   form ellipses, whereas the hyperbolic trigonometric identity

 

shows that curves of constant   form hyperbolae.

Scale factors

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In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates   are equal to

 

Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as

 

Consequently, an infinitesimal element of area equals

 

and the Laplacian reads

 

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.

Alternative definition

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An alternative and geometrically intuitive set of elliptic coordinates   are sometimes used, where   and  . Hence, the curves of constant   are ellipses, whereas the curves of constant   are hyperbolae. The coordinate   must belong to the interval [-1, 1], whereas the   coordinate must be greater than or equal to one.

The coordinates   have a simple relation to the distances to the foci   and  . For any point in the plane, the sum   of its distances to the foci equals  , whereas their difference   equals  . Thus, the distance to   is  , whereas the distance to   is  . (Recall that   and   are located at   and  , respectively.)

A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates  , so the conversion to Cartesian coordinates is not a function, but a multifunction.

 
 

Alternative scale factors

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The scale factors for the alternative elliptic coordinates   are

 
 

Hence, the infinitesimal area element becomes

 

and the Laplacian equals

 

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.

Extrapolation to higher dimensions

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Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates:

  1. The elliptic cylindrical coordinates are produced by projecting in the  -direction.
  2. The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the  -axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the  -axis, i.e., the axis separating the foci.
  3. Ellipsoidal coordinates are a formal extension of elliptic coordinates into 3-dimensions, which is based on confocal ellipsoids, hyperboloids of one and two sheets.

Note that (ellipsoidal) Geographic coordinate system is a different concept from above.

Applications

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The classic applications of elliptic coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic coordinates are a natural description of a system thus allowing a separation of variables in the partial differential equations. Some traditional examples are solving systems such as electrons orbiting a molecule or planetary orbits that have an elliptical shape.

The geometric properties of elliptic coordinates can also be useful. A typical example might involve an integration over all pairs of vectors   and   that sum to a fixed vector  , where the integrand was a function of the vector lengths   and  . (In such a case, one would position   between the two foci and aligned with the  -axis, i.e.,  .) For concreteness,  ,   and   could represent the momenta of a particle and its decomposition products, respectively, and the integrand might involve the kinetic energies of the products (which are proportional to the squared lengths of the momenta).

See also

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References

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  • "Elliptic coordinates", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill.
  • Weisstein, Eric W. "Elliptic Cylindrical Coordinates." From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html