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 -direction. The two lines of foci and of the projected Apollonian circles are generally taken to be defined by and , respectively, (and 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 is
where the coordinate of a point equals the angle and the coordinate equals the natural logarithm of the ratio of the distances and to the focal lines
(Recall that the focal lines and are located at and , respectively.)
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 -axis (the direction of projection). In the plane, the centers of the constant- and constant- cylinders lie on the and axes, respectively.
The scale factors for the bipolar coordinates and are equal
whereas the remaining scale factor . Thus, the infinitesimal volume element equals
and the Laplacian is given by
Other differential operators such as and can be expressed in the coordinates 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. A typical example would be the electric field surrounding two parallel cylindrical conductors.