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Ellipsoidal Harmonics by George Dassios

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1The ellipsoidal system and its geometry

1.1 Confocal families of second-degree surfaces

In Cartesian coordinates, a point is specified by the intersection of three planes, that is, of three first-degree surfaces. In almost all other coordinate systems, a point is specified by a combination of first- and second-degree surfaces. The ellipsoidal coordinate system is characterized by the fact that it specifies a point by using solely non-degenerate second-degree surfaces. There are three non-degenerate second- degree surfaces, that of an ellipsoid, a hyperboloid of one sheet, and a hyperboloid of two sheets, and these three surfaces are used to define the coordinate surfaces of the ellipsoidal system.

A second-degree surface in three-dimensional ...

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