Two-center bipolar coordinates

In mathematics, two-center bipolar coordinates is a coordinate system based on two coordinates which give distances from two fixed centers ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$.^[1] This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).^[2][3]

When the centers are at ${\displaystyle (+a,0)}$ and ${\displaystyle (-a,0)}$, the transformation to Cartesian coordinates${\displaystyle (x,y)}$ from two-center bipolar coordinates ${\displaystyle (r_{1},r_{2})}$ is

${\displaystyle x={\frac {r_{2}^{2}-r_{1}^{2}}{4a}}}$

${\displaystyle y=\pm {\frac {1}{4a}}{\sqrt {16a^{2}r_{2}^{2}-(r_{2}^{2}-r_{1}^{2}+4a^{2})^{2}}}}$^[1]

When x > 0, the transformation to polar coordinates from two-center bipolar coordinates is

${\displaystyle r={\sqrt {\frac {r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}}}$

${\displaystyle \theta =\arctan \left({\frac {\sqrt {r_{1}^{4}-8a^{2}r_{1}^{2}-2r_{1}^{2}r_{2}^{2}-(4a^{2}-r_{2}^{2})^{2}}}{r_{2}^{2}-r_{1}^{2}}}\right)}$

where ${\displaystyle 2a}$ is the distance between the poles (coordinate system centers).

Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.