Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system-direction. The two lines of foci of the projected Apollonian circles) 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

is

coordinate equals the natural logarithm to the focal lines

, respectively.)

correspond to cylinders of different radii

are non-intersecting cylinders of different radii

axes, respectively.

Scale factors

are equal

. Thus, the infinitesimal volume element equals

and the Laplacian is given by

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

External links

• MathWorld description of bipolar cylindrical coordinates