Editing Bipolar coordinates (section)

== Definition ==
The system is based on two [[Focus (geometry)|foci]] ''F''<sub>1</sub> and ''F''<sub>2</sub>. Referring to the figure at right, the ''σ''-coordinate of a point ''P'' equals the angle ''F''<sub>1</sub>&nbsp;''P''&nbsp;''F''<sub>2</sub>, and the ''τ''-coordinate equals the [[natural logarithm]] of the ratio of the distances ''d''<sub>1</sub> and ''d''<sub>2</sub>:

:<math>
\tau = \ln \frac{d_1}{d_2}.
</math>

If, in the Cartesian system, the foci are taken to lie at (−''a'',&nbsp;0) and (''a'',&nbsp;0), the coordinates of the point ''P'' are

:<math>
x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma}, \qquad y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma}.
</math>

The coordinate ''τ'' ranges from <math>-\infty</math> (for points close to ''F''<sub>1</sub>) to <math>\infty</math> (for points close to ''F''<sub>2</sub>). The coordinate ''σ'' is only defined modulo ''2π'', and is best taken to range from −''π'' to ''π'', by taking it as the negative of the acute angle ''F''<sub>1</sub>&nbsp;''P''&nbsp;''F''<sub>2</sub> if ''P'' is in the lower half plane.