- Elliptic cylindrical coordinates
Elliptic cylindrical coordinates are a three-dimensional orthogonal
coordinate system that results from projecting the two-dimensional elliptic coordinate system in theperpendicular -direction. Hence, thecoordinate surface s are prisms of 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 cylindrical coordinates is
:
:
:
where is a nonnegative real number and .
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 cylindrical coordinates and are equal
:
whereas the remaining scale factor . Consequently, an infinitesimal volume element 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 (x,y) 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
:
:
and, of course, . Hence, the infinitesimal volume 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 .Applications
The classic applications of elliptic cylindrical coordinates are in solving
partial differential equations , e.g.,Laplace's equation or theHelmholtz equation , for which elliptic cylindrical coordinates allow aseparation of variables . A typical example would be theelectric field surrounding a flat conducting plate of width .The three-dimensional
wave equation , when expressed in elliptic cylindrical coordinates, may be solved by separation of variables, leading to theMathieu differential equation s.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
Bibliography
* | pages = p. 657
* | pages = pp. 182–183
*, ASIN B0000CKZX7 | pages = p. 179
* | pages = p. 97
* Same as Morse & Feshbach (1953), substituting "u""k" for ξ"k".
*
External links
* [http://mathworld.wolfram.com/EllipticCylindricalCoordinates.html MathWorld description of elliptic cylindrical coordinates]
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