two-center bipolar coordinates

When the centers are at $(+a,\; 0)$ and $(-a,\; 0)$, the transformation to Cartesian coordinates $(x,\; y)$ from two-center bipolar coordinates $(r\_1,\; r\_2)$ is

:$x\; =\; \backslash frac\{r\_2^2-r\_1^2\}\{4a\}$

:$y\; =\; \backslash pm\; \backslash frac\{1\}\{4a\}\backslash sqrt\{16a^2r\_2^2-(r\_2^2-r\_1^2+4a^2)^2\}$

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

:$r\; =\; \backslash sqrt\{\backslash frac\{r\_1^2+r\_2^2-2a^2\}\{2\}\}$

:$\backslash theta\; =\; \backslash arctan\backslash left(\; \backslash frac\{\backslash sqrt\{r\_1^4-8a^2r\_1^2-2r\_1^2r\_2^2-(4a^2-r\_2^2)^2\}\}\{r\_2^2-r\_1^2\}\; \backslash right)$

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

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

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

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{{Orthogonal coordinate systems}}

Two-center bipolar coordinates

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