- Elliptic coordinates
Elliptic coordinates are a two-dimensional orthogonal
coordinate system in which thecoordinate line s are confocalellipse s andhyperbola e. The two foci and are generally taken to be fixed at and, respectively, on the -axis of theCartesian coordinate system .Basic definition
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
ellipse s, whereas the hyperbolic trigonometric identity:
shows that curves of constant form
hyperbola e.cale factors
The scale factors for the elliptic coordinates are equal
:
To simplify the computation of the scale factors, double angle identities can be used to express them equivalently as
:
Consequently, an infinitesimal element of area equals
:
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 .Alternative definition
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 they do not have a 1-to-1 transformation to the
Cartesian coordinates :
:
Alternative scale factors
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
Elliptic coordinates form the basis for several sets of three-dimensional
orthogonal coordinates . Theelliptic cylindrical coordinates are produced by projecting in the -direction.Theprolate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis connecting the foci, whereas theoblate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis separating the foci.Applications
The classic applications of elliptic coordinates are in solving
partial differential equations , e.g.,Laplace's equation or theHelmholtz equation , for which elliptic coordinates allow aseparation of variables . A typical example would be theelectric field surrounding a flat conducting plate of width .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).
ee also
*
Curvilinear coordinates
*Generalized coordinates References
* Korn GA and Korn TM. (1961) "Mathematical Handbook for Scientists and Engineers", McGraw-Hill.
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