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
 * $$x = \frac{r_2^2-r_1^2}{4a}$$


 * $$y = \pm \frac{1}{4a}\sqrt{16a^2r_2^2-(r_2^2-r_1^2+4a^2)^2}$$

Transformation to polar coordinates
When x > 0, the transformation to polar coordinates from two-center bipolar coordinates is
 * $$r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}$$


 * $$\theta = \arctan\left( \frac{\sqrt{r_1^4-8a^2r_1^2-2r_1^2r_2^2-(4a^2-r_2^2)^2}}{r_2^2-r_1^2} \right)$$

where $$2 a$$ 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.