Bipolar coordinates

Bipolar coordinates are a two-dimensional orthogonal coordinate system. There are two commonly defined types of bipolar coordinates. The first is based on the Apollonian circles. The curves of constant σ and of τ are circles that intersect at right angles. The coordinates have two foci F₁ and F₂, which are generally taken to be fixed at (−a, 0) and (a, 0), respectively, on the x-axis of a Cartesian coordinate system. The second system is two-center bipolar coordinates. There is also a third coordinate system that is based on two poles (biangular coordinates).

The term "bipolar" is sometimes used to describe other curves having two singular points (foci), such as ellipses, hyperbolas, and Cassini ovals. However, the term bipolar coordinates is reserved for the coordinates described here, and never used to describe coordinates associated with those other curves, such as elliptic coordinates.

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

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

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

The curves of constant σ correspond to non-concentric circles

that intersect at the two foci. The centers of the constant-σ circles lie on the y-axis. Circles of positive σ are centered above the x-axis, whereas those of negative σ lie below the axis. As the magnitude |σ| increases, the radius of the circles decreases and the center approaches the origin (0, 0), which is reached when |σ| = π/2, its maximum value.

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