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.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 fromThe 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.The equations for x and y can be combined to give or 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., it is an orthogonal coordinate system.As the magnitude of τ increases, the radius of the circles decreases and their centers approach the foci.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 σ and τ are equal, and given by Many results now follow in quick succession from the general formulae for orthogonal coordinates.can be expressed obtained by substituting the scale factors into the general formulae found in orthogonal coordinates.An example is the electric field surrounding two parallel cylindrical conductors with unequal diameters.
Bipolar coordinate system
Geometric interpretation of the bipolar coordinates. The angle σ is formed by the two foci and the point P , whereas τ is the logarithm of the ratio of distances to the foci. The corresponding circles of constant σ and τ are shown in red and blue, respectively, and meet at right angles (magenta box); they are orthogonal.
Two-center bipolar coordinatesBiangular coordinatesorthogonalcoordinate systemApollonian circlesellipseshyperbolasCassini ovalselliptic coordinatesnatural logarithmconformal mappingCauchy-Riemann equationsorthogonal coordinate systemorthogonal coordinatesLaplacianpartial differential equationsLaplace's equationHelmholtz equationelectric fieldbipolar cylindrical coordinatesbispherical coordinatestoroidal coordinatesElliptic coordinate systemEncyclopedia of MathematicsEMS PressKorn TMOrthogonal coordinate systemsCartesianLog-polarParabolicBipolarEllipticCylindricalSphericalParaboloidalOblate spheroidalProlate spheroidalEllipsoidalElliptic cylindricalToroidalBisphericalBipolar cylindricalConical6-sphere