Two-center bipolar coordinates

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 $c_{1}$ and $c_{2}$. This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).

Transformation to Cartesian coordinates

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

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

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

Transformation to polar coordinates

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

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

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

where $2a$ is the distance between the poles (coordinate system centers).

Applications

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

See also

• Bipolar coordinates
• Biangular coordinates
• Lemniscate of Bernoulli
• Oval of Cassini
• Cartesian oval
• Ellipse