Two-center bipolar coordinates

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, C₁ and C₂ ^[1] This system is very useful in some scientific applications^[2]^[3] It should not be confused with so-called bipolar coordinates. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and two-center bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal Coordinate system.

Cartesian coordinates

Cartesian coordinates and polar coordinates.

Transformation to Cartesian coordinates $(x,\ y)$ from two-center bipolar coordinates $(r_1,\ r_2)$ ^[1]

$x = \frac{r_1^2-r_2^2}{4a}$

$y = \pm \frac{1}{4a}\sqrt{16a^2r_1^2-(r_1^2-r_2^2+4a^2)^2}$

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

Polar coordinates

To polar coordinates from two-center bipolar coordinates

$r = \sqrt{\frac{r_1^2+r_2^2-2a^2}{2}}$

$\theta = \arctan \left[ \frac{\sqrt{8a^2(r_1^2+r_2^2 - 2a^2)-(r_1^2 - r_2^2)^2}}{r_1^2 - r_2^2}\right]\,\!$

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

References

^ ^a ^b Eric W. Weisstein, Bipolar coordinates at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
^ R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.
^ The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.

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