Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. There are also other systems, based on two poles (biangular coordinates, two-center bipolar coordinates).
The term "bipolar" is further used on occasion 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 for systems associated with those other curves, such as elliptic coordinates.
The system is based on two foci F<sub>1</sub> and F<sub>2</sub>. Referring to the figure at right, the ÃÂ-coordinate of a point P equals the angle F<sub>1</sub> P F<sub>2</sub>, and the ÃÂ-coordinate equals the natural logarithm of the ratio of the distances d<sub>1</sub> and d<sub>2</sub>:
If, in the Cartesian system, the foci are taken to lie at (âÂÂa, 0) and (a, 0), the coordinates of the point P are
The coordinate àranges from (for points close to F<sub>1</sub>) to (for points close to F<sub>2</sub>). The coordinate àis only defined modulo 2ÃÂ, and is best taken to range from âÂÂàto ÃÂ, by taking it as the negative of the acute angle F<sub>1</sub> P F<sub>2</sub> if P is in the lower half plane.
The equations for x and y can be combined to give
or
This equation shows that ÃÂ and ÃÂ are the real and imaginary parts of an analytic function of x+iy (with logarithmic branch points at the foci), which in turn proves (by appeal to the general theory of conformal mapping) (the Cauchy-Riemann equations) that these particular curves of ÃÂ and ÃÂ intersect at right angles, i.e., it is an orthogonal coordinate system.
The curves of constant ÃÂ correspond to non-concentric circles
that intersect at the two foci. The centers of the constant-àcircles lie on the y-axis at with radius . Circles of positive àare centered above the x-axis, whereas those of negative àlie below the axis. As the magnitude |ÃÂ| â ÃÂ/2 decreases, the radius of the circles decreases and the center approaches the origin (0, 0), which is reached when |ÃÂ| = ÃÂ/2. (From elementary geometry, all triangles on a circle with 2 vertices on opposite ends of a diameter are right triangles.)
The curves of constant are non-intersecting circles of different radii
that surround the foci but again are not concentric. The centers of the constant-ÃÂ circles lie on the x-axis at with radius . The circles of positive ÃÂ lie in the right-hand side of the plane (x > 0), whereas the circles of negative ÃÂ lie in the left-hand side of the plane (x < 0). The ÃÂ = 0 curve corresponds to the y-axis (x = 0). As the magnitude of ÃÂ increases, the radius of the circles decreases and their centers approach the foci.
The passage from the Cartesian coordinates towards the bipolar coordinates can be done via the following formulas:
and
The coordinates also have the identities:
and
which can derived by solving Eq. (1) and (2) for and , respectively.
To obtain the scale factors for bipolar coordinates, we take the differential of the equation for , which gives
Multiplying this equation with its complex conjugate yields
Employing the trigonometric identities for products of sines and cosines, we obtain
from which it follows that
Hence the scale factors for ÃÂ and ÃÂ are equal, and given by
Many results now follow in quick succession from the general formulae for orthogonal coordinates. Thus, the infinitesimal area element equals
and the Laplacian is given by
Expressions for , , and can be expressed obtained by substituting the scale factors into the general formulae found in orthogonal coordinates.
The classic applications of bipolar coordinates are in solving partial differential equations, e.g., Laplace's equation or the Helmholtz equation, for which bipolar coordinates allow a separation of variables. An example is the electric field surrounding two parallel cylindrical conductors with unequal diameters.
Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates.