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

of the projected Apollonian circles are generally taken to be defined by

) 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.

The most common definition of bipolar cylindrical coordinates

( σ , τ , z )

coordinate equals the natural logarithm of the ratio of the distances

to the focal lines (Recall that the focal lines

Surfaces of constant

correspond to cylinders of different radii that all pass through the focal lines and are not concentric.

The surfaces of constant

are non-intersecting cylinders of different radii that surround the focal lines but again are not concentric.

The focal lines and all these cylinders are parallel to the

plane, the centers of the constant-

The scale factors for the bipolar coordinates

are equal whereas the remaining scale factor

Thus, the infinitesimal volume element equals and the Laplacian is given by Other differential operators such as

( σ , τ )

by substituting the scale factors into the general formulae found in orthogonal coordinates.

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.