Two-center bipolar coordinates

Transformation to polar coordinates

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).

Applications

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