again are not concentric. The centers of the constant-τ circles lie on the x-axis. The circles of positive τ lie in the right-hand side of the plane (x > 0), whereas the circles of negative τ lie in the left-hand side of the plane (x < 0). The τ = 0 curve corresponds to the y-axis (x = 0). As the magnitude of τ increases, the radius of the circles decreases and their centers approach the foci. Reciprocal relations 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 is the limit one would get a x = 0 from the definition in the section above. And all the limits look quite ordinary at x =0. Scale factors 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. 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. Applications 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. Extension to 3-dimensions Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates. The bipolar cylindrical coordinates are produced by translating the bipolar coordinates along the z-axis, i.e., the out-of-plane axis. The bispherical coordinates are produced by rotating the bipolar coordinates about the x-axis, i.e., the axis connecting the foci. The toroidal coordinates are produced by rotating the bipolar coordinates about the y-axis, i.e., the axis separating the foci. References Korn GA and Korn TM. (1961) Mathematical Handbook for Scientists and Engineers, McGraw-Hill. Two-dimensional coordinate systems Orthogonal coordinate systems