Elliptic cylindrical coordinates

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction. Hence, the coordinate surfaces are prisms of 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.

Coordinate surfaces of elliptic cylindrical coordinates. The yellow sheet is the prism of a half-hyperbola corresponding to ν=-45°, whereas the red tube is an elliptical prism corresponding to μ=1. The blue sheet corresponds to z=1. The three surfaces intersect at the point P (shown as a black sphere) with Cartesian coordinates roughly (2.182, -1.661, 1.0). The foci of the ellipse and hyperbola lie at x = ±2.0.

Basic definition

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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 ellipses, whereas the hyperbolic trigonometric identity

 

shows that curves of constant   form hyperbolae.

Scale factors

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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

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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 (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

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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

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The classic applications of elliptic cylindrical coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which elliptic cylindrical coordinates allow a separation of variables. A typical example would be the electric 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 the Mathieu differential equations.

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).

Bibliography

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  • Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 657. ISBN 0-07-043316-X. LCCN 52011515.
  • Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 182–183. LCCN 55010911.
  • Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 179. LCCN 59014456. ASIN B0000CKZX7.
  • Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. p. 97. LCCN 67025285.
  • Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9. Same as Morse & Feshbach (1953), substituting uk for ξk.
  • Moon P, Spencer DE (1988). "Elliptic-Cylinder Coordinates (η, ψ, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer-Verlag. pp. 17–20 (Table 1.03). ISBN 978-0-387-18430-2.
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