- Bipolar cylindrical coordinates
**Bipolar cylindrical coordinates**are a three-dimensional orthogonalcoordinate system that results from projecting the two-dimensional bipolar coordinate system in theperpendicular $z$-direction. The two lines of foci $F\_\{1\}$ and $F\_\{2\}$ of the projectedApollonian circles are generally taken to be defined by $x=-a$ and $x=+a$, respectively, (and by $y=0$) in theCartesian coordinate system .The term "bipolar" is often used to describe other curves having two singular points (foci), such as

ellipse s,hyperbola s, andCassini oval s. However, the term "bipolar coordinates" is never used to describe coordinates associated with those curves, e.g.,elliptic coordinates .**Basic definition**The most common definition of bipolar cylindrical coordinates $(sigma,\; au,\; z)$ is

:$x\; =\; a\; frac\{sinh\; au\}\{cosh\; au\; -\; cos\; sigma\}$

:$y\; =\; a\; frac\{sin\; sigma\}\{cosh\; au\; -\; cos\; sigma\}$

:$z\; =\; z$

where the $sigma$ coordinate of a point $P$equals the angle $F\_\{1\}\; P\; F\_\{2\}$ and the $au$ coordinate equals the

natural logarithm of the ratio of the distances $d\_\{1\}$ and $d\_\{2\}$ to the focal lines:$au\; =\; ln\; frac\{d\_\{1\{d\_\{2$

(Recall that the focal lines $F\_\{1\}$ and $F\_\{2\}$ are located at $x=-a$ and $x=+a$, respectively.)

Surfaces of constant $sigma$ correspond to cylinders of different radii

:$x^\{2\}\; +left(\; y\; -\; a\; cot\; sigma\; ight)^\{2\}\; =\; frac\{a^\{2\{sin^\{2\}\; sigma\}$

that all pass through the focal lines and are not concentric. The surfaces of constant $au$ are non-intersecting cylinders of different radii

:$y^\{2\}\; +left(\; x\; -\; a\; coth\; au\; ight)^\{2\}\; =\; frac\{a^\{2\{sinh^\{2\}\; au\}$

that surround the focal lines but again are not concentric. The focal lines and all these cylinders are parallel to the $z$-axis (the direction of projection). In the $z=0$ plane, the centers of the constant-$sigma$ and constant-$au$ cylinders lie on the $y$ and $x$ axes, respectively.

**cale factors**The scale factors for the bipolar coordinates $sigma$ and $au$ are equal

:$h\_\{sigma\}\; =\; h\_\{\; au\}\; =\; frac\{a\}\{cosh\; au\; -\; cossigma\}$

whereas the remaining scale factor $h\_\{z\}=1$. Thus, the infinitesimal volume element equals

:$dV\; =\; frac\{a^\{2\{left(\; cosh\; au\; -\; cossigma\; ight)^\{2\; dsigma\; d\; au\; dz$

and the Laplacian is given by

:$abla^\{2\}\; Phi\; =frac\{1\}\{a^\{2\; left(\; cosh\; au\; -\; cossigma\; ight)^\{2\}left(\; frac\{partial^\{2\}\; Phi\}\{partial\; sigma^\{2\; +\; frac\{partial^\{2\}\; Phi\}\{partial\; au^\{2\; ight)\; +\; frac\{partial^\{2\}\; Phi\}\{partial\; z^\{2$

Other differential operators such as $abla\; cdot\; mathbf\{F\}$ and $abla\; imes\; mathbf\{F\}$ can be expressed in the coordinates $(sigma,\; au)$ by substituting the scale factors into the general formulae found in

orthogonal coordinates .**Applications**The classic applications of bipolar coordinates are in solving

partial differential equations , e.g.,Laplace's equation or theHelmholtz equation , for which bipolar coordinates allow aseparation of variables . A typical example would be theelectric field surrounding two parallel cylindrical conductors.**See also****Bibliography*** | pages = pp. 187–190

*, ASIN B0000CKZX7 | pages = p. 182

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**External links*** [

*http://mathworld.wolfram.com/BipolarCylindricalCoordinates.html MathWorld description of bipolar cylindrical coordinates*]

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