Bipolar cylindrical coordinates



Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular $$z$$-direction. The two lines of foci $$F_{1}$$ and $$F_{2}$$ of the projected Apollonian circles are generally taken to be defined by $$x=-a$$ and $$x=+a$$, respectively, (and by $$y=0$$) in the Cartesian coordinate system.

The term "bipolar" is often used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. 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 $$(\sigma, \tau, z)$$ 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.