Two-center_bipolar_coordinates Knowpia

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 edit

When the centers are at $(+a,0)$ and $(-a,0)$ , the transformation to Cartesian coordinates $(x,y)$ from two-center bipolar coordinates $(r_{1},r_{2})$ 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}}}

^[1]

Transformation to polar coordinates edit

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).

Applications edit

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

References edit

^ ^a ^b Weisstein, Eric W. "Bipolar coordinates". MathWorld.
^ 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.

[bip-1] Weisstein, Eric W. "Bipolar coordinates". MathWorld.

[2] R. Price, The Periodic Standing Wave Approximation: Adapted coordinates and spectral methods.

[3] The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.

[1]

[2]

[3]

Summary

Transformation to Cartesian coordinates edit

Transformation to polar coordinates edit

Applications edit

See also edit

References edit