Bipolar coordinates are a two-dimensionalorthogonalcoordinate system based on theApollonian circles.^[1] There is also a third system, based on two poles (biangular coordinates).

The term "bipolar" is further used on occasion to describe other curves having two singular points (foci), such asellipses,hyperbolas, andCassini ovals. However, the termbipolar coordinates is reserved for the coordinates described here, and never used for systems associated with those other curves, such aselliptic coordinates.

The system is based on twofociF₁ andF₂. Referring to the figure at right, theσ-coordinate of a pointP equals the angleF₁ P F₂, and theτ-coordinate equals thenatural logarithm of the ratio of the distancesd₁ andd₂:

${\displaystyle \tau =\ln \frac{d_1}{d_2}.}$

If, in the Cartesian system, the foci are taken to lie at (−a, 0) and (a, 0), the coordinates of the pointP are

${\displaystyle x=a\frac{\sinh \tau }{\cosh \tau -\cos \sigma },y=a\frac{\sin \sigma }{\cosh \tau -\cos \sigma }.}$

The coordinateτ ranges from${\displaystyle -\mathrm{\infty }}$ (for points close toF₁) to${\displaystyle \mathrm{\infty }}$ (for points close toF₂). The coordinateσ is only defined modulo2π, and is best taken to range from −π toπ, by taking it as the negative of the acute angleF₁ P F₂ ifP is in the lower half plane.

The equations forx andy can be combined to give

${\displaystyle x+iy=ai\cot \left(\frac{\sigma +i\tau }{2}\right)}$^[2][3]

or

${\displaystyle x+iy=a\coth \left(\frac{\tau -i\sigma }{2}\right).}$

This equation shows thatσ andτ are the real and imaginary parts of ananalytic function ofx+iy (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory ofconformal mapping) (theCauchy-Riemann equations) that these particular curves ofσ andτ intersect at right angles, i.e., it is anorthogonal coordinate system.

The curves of constantσ correspond to non-concentric circles

${\displaystyle x^2+{\left(y-a\cot \sigma \right)}^2=\frac{a^2}{{\sin }^2\sigma }=a^2(1+{\cot }^2\sigma )}$

(1)

that intersect at the two foci. The centers of the constant-σ circles lie on they-axis at${\displaystyle a\cot \sigma }$ with radius${\displaystyle \frac{a}{\sin \sigma }}$. Circles of positiveσ are centered above thex-axis, whereas those of negativeσ lie below the axis. As the magnitude |σ| −π/2 decreases, the radius of the circles decreases and the center approaches the origin (0, 0), which is reached when |σ| =π/2. (From elementary geometry, all triangles on a circle with 2 vertices on opposite ends of a diameter are right triangles.)

The curves of constant${\displaystyle \tau }$ are non-intersecting circles of different radii

${\displaystyle {\left(x-a\coth \tau \right)}^2+y^2=\frac{a^2}{{\sinh }^2\tau }=a^2({\coth }^2\tau -1)}$

(2)

that surround the foci but again are not concentric. The centers of the constant-τ circles lie on thex-axis at${\displaystyle a\coth \tau }$ with radius${\displaystyle \frac{a}{\sinh \tau }}$. The circles of positiveτ lie in the right-hand side of the plane (x > 0), whereas the circles of negativeτ lie in the left-hand side of the plane (x < 0). Theτ = 0 curve corresponds to they-axis (x = 0). As the magnitude ofτ increases, the radius of the circles decreases and their centers approach the foci.

The passage from the Cartesian coordinates towards the bipolar coordinates can be done via the following formulas:

${\displaystyle \tau =\frac{1}{2}\ln \frac{(x+a{)}^2+y^2}{(x-a{)}^2+y^2}}$

and

${\displaystyle \pi -\sigma =2\arctan \frac{2ay}{a^2-x^2-y^2+\sqrt{(a^2-x^2-y^2{)}^2+4a^2{y}^2}}.}$

The coordinates also have the identities:

${\displaystyle \tanh \tau =\frac{2ax}{x^2+y^2+a^2}}$

and

${\displaystyle \tan \sigma =\frac{2ay}{x^2+y^2-a^2},}$

which can derived by solving Eq. (1) and (2) for${\displaystyle \cot \sigma }$ and${\displaystyle \coth \tau }$, respectively.

To obtain the scale factors for bipolar coordinates, we take the differential of the equation for${\displaystyle x+iy}$, which gives

${\displaystyle dx+idy=\frac{-ia}{{\sin }^2(\frac{1}{2}(\sigma +i\tau ))}(d\sigma +id\tau ).}$

Multiplying this equation with itscomplex conjugate yields

${\displaystyle (dx{)}^2+(dy{)}^2=\frac{a^2}{[2\sin \frac{1}{2}(\sigma +i\tau )\sin \frac{1}{2}(\sigma -i\tau ){]}^2}((d\sigma {)}^2+(d\tau {)}^2).}$

Employing the trigonometric identities for products of sines and cosines, we obtain

${\displaystyle 2\sin \frac{1}{2}(\sigma +i\tau )\sin \frac{1}{2}(\sigma -i\tau )=\cos \sigma -\cosh \tau ,}$

from which it follows that

${\displaystyle (dx{)}^2+(dy{)}^2=\frac{a^2}{(\cosh \tau -\cos \sigma {)}^2}((d\sigma {)}^2+(d\tau {)}^2).}$

Hence the scale factors forσ andτ are equal, and given by

${\displaystyle h_{\sigma }=h_{\tau }=\frac{a}{\cosh \tau -\cos \sigma }.}$

Many results now follow in quick succession from the general formulae fororthogonal coordinates.Thus, theinfinitesimal area element equals

${\displaystyle dA=\frac{a^2}{{\left(\cosh \tau -\cos \sigma \right)}^2}d\sigma d\tau ,}$

and theLaplacian is given by

${\displaystyle {\mathrm{\nabla }}^2\mathrm{\Phi }=\frac{1}{a^2}{\left(\cosh \tau -\cos \sigma \right)}^2\left(\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{\sigma }^2}+\frac{{\mathrm{\partial }}^2\mathrm{\Phi }}{\mathrm{\partial }{\tau }^2}\right).}$

Expressions for${\displaystyle \mathrm{\nabla }f}$,${\displaystyle \mathrm{\nabla }\cdot \mathbf{F}}$, and${\displaystyle \mathrm{\nabla }\times \mathbf{F}}$ can be expressed obtained by substituting the scale factors into the general formulae found inorthogonal coordinates.

The classic applications of bipolar coordinates are in solvingpartial differential equations, e.g.,Laplace's equation or theHelmholtz equation, for which bipolar coordinates allow aseparation of variables. An example is theelectric field surrounding two parallel cylindrical conductors with unequal diameters.

Bipolar coordinates form the basis for several sets of three-dimensionalorthogonal coordinates.

• Thebipolar cylindrical coordinates are produced by translating the bipolar coordinates along thez-axis, i.e., the out-of-plane axis.

• Thebispherical coordinates are produced by rotating the bipolar coordinates about thex-axis, i.e., the axis connecting the foci.

• Thetoroidal coordinates are produced by rotating the bipolar coordinates about they-axis, i.e., the axis separating the foci.

• Elliptic coordinate system

• Interactive demo with desmoshttps://www.desmos.com/calculator/nbvnucu4o5

• "Bipolar coordinates",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Korn GA andKorn TM. (1961)Mathematical Handbook for Scientists and Engineers, McGraw-Hill.

• Two-dimensional coordinate systems
• Orthogonal coordinate systems

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• Short description matches Wikidata