Elliptic coordinate system

In an elliptical coordinate system, a point in the plane is determined by specifying the location on confocal ellipses and hyperbolas. More generally, there are also elliptical coordinate systems in three-dimensional space.

Plane elliptic coordinates

Definition

Usually to select the two focal points at the positions and on the x- axis of a Cartesian coordinate system. The point with the elliptic coordinates then the Cartesian coordinates

With and. Summing up the plane as a complex plane, it shall

Properties

The coordinate lines are hyperbolas coordinates lines ellipses. For the coordinate line is degenerate to link the two focal points. For the coordinate line is degenerate to the half-lines on the x- axis, for to this mirror-symmetric half-line on the negative x -axis. And the coordinates for the line is positive and the negative y -axis.

All ellipses and hyperbolas have the same linear eccentricity e = c. The ellipses on which is constant, have the semi-major axis, the semi-minor axis and eccentricity. The hyperbolas on which is constant, have the real semi-axis, the imaginary semi-axis and eccentricity.

Generalization to three dimensions

This ellipsoidal coordinates may be extended to different kinds of three-dimensional space. For cylindrical elliptical coordinates is added as an additional coordinate simply the Cartesian z -coordinate. In polar coordinates, the elliptic plane is rotated by an angle, which then forms the additional coordinate:

Finally, there are spatially elliptic coordinates:

Here, b is a further parameter of the coordinate system. The coordinate lines are ellipses here. The coordinate runs from 0 to here, the coordinate from 0 to infinity and from 0 to.

Applications

Through the transformation to elliptic coordinates the Schrödinger equation for the H2 can - molecule is dissolved in Born- Oppenheimer approximation analytically.

• Geometry