Bipolar cylindrical coordinates

Basic definition

The most common definition of bipolar cylindrical coordinates (\sigma, \tau, z) is

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

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

: z = \ z

where the \sigma coordinate of a point P equals the angle F_{1} P F_{2} and the \tau coordinate equals the natural logarithm of the ratio of the distances d_{1} and d_{2} to the focal lines

: \tau = \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 \right)^{2} = \frac{a^{2}}{\sin^{2} \sigma}

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

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

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-\tau cylinders lie on the y and x axes, respectively.

Scale factors

The scale factors for the bipolar coordinates \sigma and \tau are equal

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

whereas the remaining scale factor h_{z}=1. Thus, the infinitesimal volume element equals

: dV = \frac{a^{2}}{\left( \cosh \tau - \cos\sigma \right)^{2}} d\sigma d\tau dz

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) + \frac{\partial^{2} \Phi}{\partial z^{2}}

Other differential operators such as \nabla \cdot \mathbf{F} and \nabla \times \mathbf{F} can be expressed in the coordinates (\sigma, \tau) 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 (in 2D). A typical example would be the electric field surrounding two parallel cylindrical conductors.

Bibliography