Alternative Definition
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, 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.
Read more about this topic: Elliptic Coordinate System
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