Elliptical coordinates

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Elliptical coordinates in the plane for c = 1. The numerical ellipticity is denoted here by e.

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

Plane elliptical coordinates

definition

Usually one chooses the two focal points at the positions and on the axis of a Cartesian coordinate system . The point with the elliptical coordinates then has the Cartesian coordinates

with and . If the level is understood as a complex level , then the following applies

Transformations

The above relationships are simply used to transform from elliptical to Cartesian coordinates .

In order to carry out the inverse transformation one has to take the basic idea of ​​these coordinates as an aid. This states that the point must lie on both an ellipse and a confocal hyperbola . These have semi-axes as indicated in the section below. With the aid of the ellipse and hyperbola equation in Cartesian coordinates and the form of the principal axis , it follows:

and

.

These equations are satisfied by the Cartesian representations given above.

From this, using the elementary relationships of the trigonometric or hyperbolic functions

and

derive the following transformation rules:

with the substitution to simplify writing .

Further transformations, such as from planar polar coordinates to elliptical coordinates, can be carried out using the intermediate step of the Cartesian coordinates.

properties

The - coordinate lines are hyperbolas that -Koordinatenlinien ellipses . For the coordinate line to the line connecting the two foci is degenerate. For the -coordinate line is degenerate to the half-line on the -axis , for to the mirror-symmetrical half-line on the negative -axis. For and the coordinate line is the positive or negative axis.

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

The representation in this coordinate form is only possible because hyperbolic cos and hyperbolic sine or cosine and sine trivially fulfill the relationships between major and minor semiaxes ( ) for ellipses and real and imaginary semiaxes for hyperbolas ( ).

Generalization to three dimensions

These elliptical coordinates can be extended to three-dimensional space in various ways. In the case of cylindrical elliptical coordinates, the Cartesian coordinate is simply added as an additional coordinate. With polar elliptical coordinates, the plane is rotated by an angle , which then forms the additional coordinate:

Finally, there are also spatially elliptical coordinates:

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

Applications

By the transformation on elliptical coordinates can Schrodinger equation for the H 2 + molecule in Born-Oppenheimer approximation be solved analytically.

literature

  • DD Sokolov: Elliptic coordinates . In: Michiel Hazewinkel (Ed.): Encyclopedia of Mathematics . Springer, 2001

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