Constant Coordinate Curves for Elliptic Coordinates

This Demonstration shows curves of constant coordinate values for the elliptic coordinate system in two dimensions. These curves are semi-hyperbolas and ellipses, the latter having foci at . As you drag the locator in the plane, the curves are redrawn so they pass through that point. Holding the mouse over the curve shows which variable is constant along that curve, and holding it over the point gives the actual values of the variables. You can vary the interfocal separation, ; at the elliptic coordinates are equivalent to polar coordinates.

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Two-dimensional elliptic coordinates may be defined by , , for and . The curves of constant and are ellipses and hyperbolas, respectively.
The inverse relation can be expressed , , where is the distance from the left/right focus. The slightly ungainly factor is necessary here to ensure that the correct (upper/lower) half-plane is chosen.
In the limit , the elliptic coordinates reduce to polar coordinates . The correspondence is given by and (note that itself becomes infinite as ).
Three-dimensional generalizations of the elliptic coordinates are the oblate and prolate spheroidal coordinates, elliptic cylindrical coordinates, and ellipsoidal coordinates.
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