- Bipolar coordinates
**Bipolar coordinates**are a two-dimensional orthogonalcoordinate system . There are two commonly defined types of bipolar coordinates. [*http://bbs.sachina.pku.edu.cn/Stat/Math_World/math/b/b233.htm Eric W. Weisstein,*] ] The other system is**Concise Encyclopedia of Mathematics CD-ROM**, "Bipolar Coordinates", CD-ROM edition 1.0, May 20, 1999two-center bipolar coordinates . There is also third coordinate system that based on two poles (biangular coordinates ). First is based on theApollonian circles . The curves of constant "σ" and of "τ" are circles that intersect at right angles. The coordinates have two foci "F"_{1}and "F"_{2}, which are generally taken to be fixed at (−"a", 0) and ("a", 0), respectively, on the "x"-axis of aCartesian coordinate system .Bipolar coordinates form the basis for several sets of three-dimensional

orthogonal coordinates . Thebipolar cylindrical coordinates are produced by projecting in the "z"-direction. Thebispherical coordinates are produced by rotating the bipolar coordinates about the $x$-axis, i.e., the axis connecting the foci, whereas thetoroidal coordinates are produced by rotating the bipolar coordinates about the "y"-axis, i.e., the axis separating the foci.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.The term "bipolar" is sometimes 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 reserved for the coordinates described here, and never used to describe coordinates associated with those other curves, such aselliptic coordinates .**Basic definition**The most common definition of bipolar coordinates ("σ", "τ") is

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

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

where the "σ"-coordinate of a point "P" equals the angle "F"

_{1}"P" "F"_{2}and the "τ"-coordinate equals thenatural logarithm of the ratio of the distances "d"_{1}and "d"_{2}to the foci:$au\; =\; ln\; frac\{d\_1\}\{d\_2\}$

(Recall that "F"

_{1}and "F"_{2}are located at (−"a", 0) and ("a", 0), respectively.)It should be noted that the "σ" and "τ" isosurfaces intersect at two points, not one. The foci divide each "σ"-isosurface circle into a longer and shorter arc. By convention, the "σ" on these two arcs differ by 90°, i.e., "σ"

_{shorter}= "σ"_{longer}+ 90°; this convention breaks the degeneracy of the coordinate system.**Curves of constant "σ" and "τ"**The curves of constant "σ" correspond to non-concentric circles

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

that intersect at the two foci. The centers of the constant-"σ" circles lie on the "y"-axis. Circles of positive "σ" are centered above the "x"-axis, whereas those of negative "σ" lie below the axis. As the magnitude |"σ"| increases, the radius of the circles decreases and the center approaches the origin (0, 0), which is reached when |"σ"| = "π"/2, its maximum value.

The curves of constant $au$ are non-intersecting circles of different radii

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

that surround the foci but again are not concentric. The centers of the constant-"τ" circles lie on the "x"-axis. The circles of positive "τ" lie in the right-hand side of the plane ("x" > 0), whereas the circles of negative "τ" lie in the left-hand side of the plane ("x" < 0). The "τ" = 0 curve corresponds to the "y"-axis ("x" = 0). As the magnitude of "τ" increases, the radius of the circles decreases and their centers approach the foci.

**cale factors**The scale factors for the bipolar coordinates ("σ", "τ") are equal

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

Thus, the infinitesimal area element equals

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

and the

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

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

orthogonal coordinates .**ee also****References*** H. Bateman "Spheroidal and bipolar coordinates", "Duke Mathematical Journal" 4 (1938), no. 1, 39–50

* Lockwood, E. H. "Bipolar Coordinates." Chapter 25 in "A Book of Curves". Cambridge, England: Cambridge University Press, pp. 186–190, 1967.

* Korn GA and Korn TM. (1961) "Mathematical Handbook for Scientists and Engineers", McGraw-Hill.**Notes**

*Wikimedia Foundation.
2010.*