Two-center bipolar coordinates

For related concepts, see Bipolar coordinates.

In mathematics, two-center bipolar coordinates is a coordinate system, based on two coordinates which give distances from two fixed centers, $c_{1}$ and $c_{2}$ .^[1] This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).^[2]^[3]

Transformation to Cartesian coordinates

Cartesian coordinates and polar coordinates.

The transformation to Cartesian coordinates $(x,\ y)$ from two-center bipolar coordinates $(r_{1},\ r_{2})$ is

x={\frac {r_{2}^{2}-r_{1}^{2}}{4a}}

y=\pm {\frac {1}{4a}}{\sqrt {16a^{2}r_{2}^{2}-(r_{2}^{2}-r_{1}^{2}+4a^{2})^{2}}}

where the centers of this coordinate system are at $(+a,\ 0)$ and $(-a,\ 0)$ .^[1]

Transformation to polar coordinates

When x>0 the transformation to polar coordinates from two-center bipolar coordinates is

r={\sqrt {{\frac {r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}}}

\theta =\arctan \left({\frac {{\sqrt {r_{1}^{4}-8a^{2}r_{1}^{2}-2r_{1}^{2}r_{2}^{2}-(4a^{2}-r_{2}^{2})^{2}}}}{r_{2}^{2}-r_{1}^{2}}}\right)\,\!

where $2a$ is the distance between the poles (coordinate system centers).

References

Orthogonal coordinate systems

Two dimensional	Cartesian coordinate system Polar coordinate system Parabolic coordinate system Bipolar coordinates Elliptic coordinates

Three dimensional	Cartesian coordinate system Cylindrical coordinate system Spherical coordinate system Parabolic cylindrical coordinates Paraboloidal coordinates Oblate spheroidal coordinates Prolate spheroidal coordinates Ellipsoidal coordinates Elliptic cylindrical coordinates Toroidal coordinates Bispherical coordinates Bipolar cylindrical coordinates Conical coordinates 6-sphere coordinates Flat-ring cyclide coordinates Flat-disk cyclide coordinates Bi-cyclide coordinates Cap-cyclide coordinates Concave bi-sinusoidal single-centered coordinates Concave bi-sinusoidal double-centered coordinates Convex inverted-sinusoidal spherically aligned coordinates Quasi-random-intersection cartesian coordinates

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Two-center bipolar coordinates

Transformation to Cartesian coordinates

Transformation to polar coordinates

See also

References