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

The transformation to Cartesian coordinates ${\displaystyle (x,\ y)}$ from two-center bipolar coordinates ${\displaystyle (r_{1},\ r_{2})}$ is

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