Elliptic coordinate system

In geometry, the elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae. The two foci and are generally taken to be fixed at and , respectively, on the -axis of the Cartesian coordinate system. The most common definition of elliptic coordinates is where is a nonnegative real number and On the complex plane, an equivalent relationship is These definitions correspond to ellipses and hyperbolae. The trigonometric identity shows that curves of constant form ellipses, whereas the hyperbolic trigonometric identity shows that curves of constant form hyperbolae. In an orthogonal coordinate system the lengths of the basis vectors are known as scale factors. The scale factors for the elliptic coordinates are equal to Using the double argument identities for hyperbolic functions and trigonometric functions, the scale factors can be equivalently expressed as Consequently, an infinitesimal element of area equals and the Laplacian reads Other differential operators such as and can be expressed in the coordinates by substituting the scale factors into the general formulae found in orthogonal coordinates. An alternative and geometrically intuitive set of elliptic coordinates are sometimes used, where and . Hence, the curves of constant are ellipses, whereas the curves of constant are hyperbolae. The coordinate must belong to the interval [-1, 1], whereas the coordinate must be greater than or equal to one. The coordinates have a simple relation to the distances to the foci and . For any point in the plane, the sum of its distances to the foci equals , whereas their difference equals . Thus, the distance to is , whereas the distance to is . (Recall that and are located at and , respectively.) A drawback of these coordinates is that the points with Cartesian coordinates (x,y) and (x,-y) have the same coordinates , so the conversion to Cartesian coordinates is not a function, but a multifunction.