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 perpendicularof 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.The most common definition of bipolar cylindrical coordinatescoordinate equals the natural logarithm of the ratio of the distancescorrespond to cylinders of different radii that all pass through the focal lines and are not concentric.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 theplane, the centers of the constant-The scale factors for the bipolar coordinatesare equal whereas the remaining scale factorThus, the infinitesimal volume element equals and the Laplacian is given by Other differential operators such asby 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.
Coordinate surfaces of the bipolar cylindrical coordinates. The yellow crescent corresponds to σ, whereas the red tube corresponds to τ and the blue plane corresponds to z =1. The three surfaces intersect at the point P (shown as a black sphere).
orthogonalcoordinate systembipolar coordinate systemApollonian circlesCartesian coordinate systemellipseshyperbolasCassini ovalselliptic coordinatesnatural logarithmorthogonal coordinatespartial differential equationsLaplace's equationHelmholtz equationseparation of variableselectric fieldMargenau HKorn TMOrthogonal coordinate systemsCartesianLog-polarParabolicBipolarEllipticCylindricalSphericalParaboloidalOblate spheroidalProlate spheroidalEllipsoidalElliptic cylindricalToroidalBisphericalConical6-sphere