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