Bipolar coordinates are a two-dimensional orthogonal coordinate system based on the Apollonian circles. Confusingly, the same term is also sometimes used for two-center bipolar coordinates. There is also a third system, based on two poles (biangular 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. Definition The system is based on two foci F1 and F2. Referring to the figure at right, 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: If, in the Cartesian system, the foci are taken to lie at (−a, 0) and (a, 0), the coordinates of the point P are The coordinate τ ranges from (for points close to F1) to (for points close to F2). 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 F1 P F2 if P is in the lower half plane. Proof that coordinate system is orthogonal The equations for x and y can be combined to give This equation shows that σ and τ 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 σ and τ intersect at right angles, i.e., that the coordinate system is orthogonal. Curves of constant σ and τ 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|- π/2 decreases, the radius of the circles decreases and the center approaches the origin (0, 0), which is reached when |σ| = π/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