Ellipsoidal coordinates
spatial elliptic coordinates
The numbers ,
and
connected with Cartesian rectangular coordinates
,
and
by the formulas
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where . The coordinate surfaces are (see Fig.): ellipses
, one-sheet hyperbolas (
), and two-sheet hyperbolas (
), with centres at the coordinate origin.
Figure: e035420a
The system of ellipsoidal coordinates is orthogonal. To every triple of numbers ,
and
correspond 8 points (one in each octant), which are symmetric to each other relative to the coordinate planes of the system
.
The Lamé coefficients are
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If one of the conditions in the definition of ellipsoidal coordinates is replaced by an equality, then degenerate ellipsoidal coordinate systems are obtained.
Comments
References
[a1] | G. Darboux, "Leçons sur la théorie générale des surfaces et ses applications géométriques du calcul infinitésimal" , 1 , Gauthier-Villars (1887) pp. 1–18 |
Ellipsoidal coordinates. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ellipsoidal_coordinates&oldid=14024