Bipolar Coordinates - Definition

Definition

The most common definition of bipolar coordinates (σ, τ) is

x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}
y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}

where the σ-coordinate of a point P equals the angle F1 P F2 and the τ-coordinate equals the natural logarithm of the ratio of the distances d1 and d2 to the foci

\tau = \ln \frac{d_1}{d_2}

(Recall that F1 and F2 are located at (−a, 0) and (a, 0), respectively.) Equivalently

x + i y = a i \cot\left( \frac{\sigma + i \tau}{2}\right)

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