Bipolar coordinates

See also: two-center bipolar coordinates

Bipolar coordinate system

Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. The first is based on the Apollonian circles. The curves of constant σ and of τ are circles that intersect at right angles. The coordinates have two foci F₁ and F₂, which are generally taken to be fixed at (−a, 0) and (a, 0), respectively, on the x-axis of a Cartesian coordinate system. The second system is two-center bipolar coordinates. There is also a third coordinate system that is based on two poles (biangular coordinates).

The term "bipolar" is sometimes used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is reserved for the coordinates described here, and never used to describe coordinates associated with those other curves, such as elliptic coordinates.

Geometric interpretation of the bipolar coordinates. The angle σ is formed by the two foci and the point P, whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.

Definition

The most common definition of bipolar coordinates (σ, τ) is

$x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}$

$y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}$

where the σ-coordinate of a point P equals the angle F₁ P F₂ and the τ-coordinate equals the natural logarithm of the ratio of the distances d₁ and d₂ to the foci

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

(Recall that F₁ and F₂ are located at (−a, 0) and (a, 0), respectively.) Equivalently

$x + i y = a i \cot\left( \frac{\sigma + i \tau}{2}\right)$

Curves of constant σ and τ

The curves of constant σ correspond to non-concentric circles

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

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 $\tau$ are non-intersecting circles of different radii

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

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.

Reciprocal relations

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

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

and

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

We notice also those two remarkable identities:

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

and

$\tan \sigma = \frac{2 a y}{x^2 + y^2 - a^2}.$

Scale factors

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

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

Thus, the infinitesimal area element equals

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

and the Laplacian is given by

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

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates (σ, τ) 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 the Helmholtz equation, for which bipolar coordinates allow a separation of variables. A typical example would be the electric field surrounding two parallel cylindrical conductors.

Extension to 3-dimensions

Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates. The bipolar cylindrical coordinates are produced by projecting in the z-direction. The bispherical coordinates are produced by rotating the bipolar coordinates about the $x$-axis, i.e., the axis connecting the foci, whereas the toroidal coordinates are produced by rotating the bipolar coordinates about the y-axis, i.e., the axis separating the foci.

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.