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There are several different ways to define a bipolar coordinate system. Two of the most common are Grubert’s bipolar system and the bipolar system of two concentric circles.

## Grubert’s Bipolar System

The **bipolar coordinate system**, created by Grubert in 1859 [1], is an unusual coordinate system where coordinates of a point are its distance from two poles, or fixed points. In other words, it’s a system with two points that are a specified distance apart, where a point’s coordinates are the its distance from the two points, A plus or minus indicates which side of the line joining the two poles the point is [2].

It is sometimes called a dipolar, biradial, vectorial, or bivectorial system.

## Bipolar Coordinate System: Concentric Circles

An **alternate definition** of a bipolar coordinate system is one where two concentric families of circles share two common center points; the bipolar coordinates of a point are the values of two parameters defining the concentric circles. The distance between the two center points, or foci, is usually a constant 2*a*, but may be variable in some cases (e.g., the van der Grinten map projection) [2].

The formula, which describes how τ and σ are connected with the Cartesian coordinates x and y, is:

Where 0 ≤ σ < π, -∞ < τ < ∞.

The bipolar coordinate system has two **rotational forms**, toroidal and bispherical [3].

## What is the Bipolar Coordinate System Used For?

The Bipolar Coordinate System has a range of applications, including providing an unambiguous localization of points in a 3D space [4]. Practical applications include [5]:

- Lines of flux for an electric dipole,
- Electric fields between two spheres,
- Parallel microwave transmission lines.

While the system is sometimes helpful, it isn’t easy to work with. Mark Wagner states in The *Geometries of Visual Space* that it is “awkward to use at best” [6].

## References

Image: Mkwadee,

[1] Wood, F. Coordinate Systems in One and Two Dimensions. Retrieved January 19, 2022 from: https://kuscholarworks.ku.edu/bitstream/handle/1808/14967/Wood_Coordinate_Systems%20(Smaller).pdf;jsessionid=2550A8539FB17CD6E492C09B5C0CCF4E?sequence=1

[2] American Society of Civil Engineers. (1994). Glossary of the Mapping Sciences.

[3] Arfken (1970). Chapter 1.

[4] Wade, G. (2012). Acoustical Holography

Volume 4 Proceedings of the Fourth International Symposium on Acoustical Holography, Held in Santa Barbara, California, April 10–12, 1972.

[5] Westgard, J. (2012). Electrodynamics: A Concise Introduction. Springer New York.

[6] Wagner, M. (2012). The Geometries of Visual Space. Taylor & Francis.

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