Two-center bipolar coordinates explained

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).^{[1] [2]}

Transformation to Cartesian coordinates

When the centers are at

( + a, 0 ) and ( - a, 0 ) , the transformation to Cartesian coordinates (x, y) from two-center bipolar coordinates (r _{1,} r _{2)} isx = (r ^{2} _{1})/(4 a) y = \pm (1)/(4 a) \sqrt{ 16 a ^{2+4 a} _{1} ^{2)} ^{2}}

Transformation to polar coordinates

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

r = \sqrt{ (r ^{2-2 a} _{2} ^{2})/(2) }\theta = \arctan\left( (\sqrt{r ^{4-8 a} _{1} ^{2- ( 4 a} ^{2} ^{2 )} ^{2} ^{2})/() } \right) where 2 a is the distance between the poles (coordinate system centers).

Applications

Polar plotters use two-center bipolar coordinates to describe the drawing paths required to draw a target image.

See also

• Bipolar coordinates
• Biangular coordinates
• Lemniscate of Bernoulli
• Oval of Cassini
• Cartesian oval
• Ellipse

References

1. http://www.physics.utah.edu/~rprice/AREA51DOCS/paperIIa.pdf R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.
2. https://arxiv.org/abs/gr-qc/0502034v1 The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.