 |thumb|right|350px|bipolar coordinate system bipolar coordinates are a two-dimensional orthogonal coordinate system. there are two commonly defined types of bipolar coordinates.eric w. weisstein, concise encyclopedia of mathematics cd-rom, bipolar coordinates, cd-rom edition 1.0, may 20, 1999 the first is based on the apollonian circles. the curves of constant Ï and of Ï are circles that intersect at right angles. the coordinates have two foci f1 and f2, which are generally taken to be fixed at (âa,&nbsp;0) and (a,&nbsp;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. thumb|right|350px|geometric interpretation of the bipolar coordinates. the angle Ï is formed by the two foci and the point p, whereas Ï is the logarithm of the ratio of distances to the foci. the corresponding circles of constant Ï and Ï are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal. == definition == the most common definition of bipolar coordinates (Ï,&nbsp;Ï) is : x = a \ \frac{\sinh \tau}{\cosh \tau - \cos \sigma} : y = a \ \frac{\sin \sigma}{\cosh \tau - \cos \sigma} where the Ï-coordinate of a point p equals the angle f1&nbsp;p&nbsp;f2 and the Ï-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 (âa,&nbsp;0) and (a,&nbsp;0), respectively.) equivalently : x + i y = a i \cot\left( \frac{\sigma + i \tau}{2}\right) == curves of constant Ï and Ï == right|280px right|280px the curves of constant Ï correspond to non-concentric circles : x^2 + \left( y - a \cot \sigma \right)^2 = \frac{a^{2}}{\sin^2 \sigma} that intersect at the two foci. the centers of the constant-Ï circles lie on the y-axis. circles of positive Ï are centered above the x-axis, whereas those of negative Ï lie below the axis. as the magnitude |Ï| increases, the radius of the circles decreases and the center approaches the or
Coordinate_systems