Bipolar cylindrical coordinates

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 ${\displaystyle (\sigma ,\tau ,z)}$ is

${\displaystyle x=a\frac{\sinh \tau }{\cosh \tau -\cos \sigma }}$

${\displaystyle y=a\frac{\sin \sigma }{\cosh \tau -\cos \sigma }}$

${\displaystyle z=z}$

where the ${\displaystyle \sigma }$ coordinate of a point ${\displaystyle P}$ equals the angle ${\displaystyle F_{1}P{F}_2}$ and the ${\displaystyle \tau }$ coordinate equals the natural logarithm of the ratio of the distances ${\displaystyle d_1}$ and ${\displaystyle d_2}$ to the focal lines

${\displaystyle \tau =\ln \frac{d_1}{d_2}}$

(Recall that the focal lines ${\displaystyle F_1}$ and ${\displaystyle F_2}$ are located at ${\displaystyle x=-a}$ and ${\displaystyle x=+a}$, respectively.)

Surfaces of constant ${\displaystyle \sigma }$ correspond to cylinders of different radii

${\displaystyle x^2+{\left(y-a\cot \sigma \right)}^2=\frac{a^2}{{\sin }^2\sigma }}$

that all pass through the focal lines and are not concentric. The surfaces of constant ${\displaystyle \tau }$ are non-intersecting cylinders of different radii

${\displaystyle y^2+{\left(x-a\coth \tau \right)}^2=\frac{a^2}{{\sinh }^2\tau }}$

that surround the focal lines but again are not concentric. The focal lines and all these cylinders are parallel to the ${\displaystyle z}$-axis (the direction of projection). In the ${\displaystyle z=0}$ plane, the centers of the constant-${\displaystyle \sigma }$ and constant-${\displaystyle \tau }$ cylinders lie on the ${\displaystyle y}$ and ${\displaystyle x}$ axes, respectively.