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Two center bipolar coordinates

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Two-center 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 . This system is very useful in some scientific applications (e.g. calculating the electric field of a dipole on a plane).

Contents

Transformation to Cartesian coordinates

The transformation to Cartesian coordinates ( x ,   y ) from two-center bipolar coordinates ( r 1 ,   r 2 ) is

x = r 2 2 − r 1 2 4 a y = ± 1 4 a 16 a 2 r 2 2 − ( r 2 2 − r 1 2 + 4 a 2 ) 2

where the centers of this coordinate system are at ( + a ,   0 ) and ( − a ,   0 ) .

Transformation to polar coordinates

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

r = r 1 2 + r 2 2 − 2 a 2 2 θ = arctan ⁡ ( r 1 4 − 8 a 2 r 1 2 − 2 r 1 2 r 2 2 − ( 4 a 2 − r 2 2 ) 2 r 2 2 − r 1 2 )

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

References

Two-center bipolar coordinates Wikipedia
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