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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, two-center bipolar coordinates is a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...
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 An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
of a dipole on a plane).The periodic standing-wave approximation: nonlinear scalar fields, adapted coordinates, and the eigenspectral method.
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Transformation to Cartesian coordinates

When the centers are at (+a, 0) and (-a, 0), the transformation to
Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
(x, y) from two-center bipolar coordinates (r_1, r_2) is :x = \frac :y = \pm \frac\sqrt


Transformation to polar coordinates

When ''x'' > 0, the transformation to
polar coordinates In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference ...
from two-center bipolar coordinates is :r = \sqrt :\theta = \arctan\left( \frac \right) where 2 a is the distance between the poles (coordinate system centers).


Applications

Polar plotter A polar plotter also known as polargraph or Kritzler is a plotter which uses two-center bipolar coordinates to produce vector drawings using a pen suspended from strings connected to two pulleys at the top of the plotting surface. This gives it tw ...
s use two-center bipolar coordinates to describe the drawing paths required to draw a target image.


See also

*
Bipolar coordinates Bipolar coordinates are a two-dimensional orthogonal coordinates, orthogonal coordinate system based on the Apollonian circles.Eric W. Weisstein, Concise Encyclopedia of Mathematics CD-ROM, ''Bipolar Coordinates'', CD-ROM edition 1.0, May 20, 19 ...
*
Biangular coordinates In mathematics, biangular coordinates are a coordinate system for the plane where C_1 and C_2 are two fixed points, and the position of a point ''P'' not on the line \overline is determined by the angles \angle PC_1C_2 and \angle PC_2C_1. The ...
*
Lemniscate of Bernoulli In geometry, the lemniscate of Bernoulli is a plane curve defined from two given points and , known as foci, at distance from each other as the locus of points so that . The curve has a shape similar to the numeral 8 and to the ∞ symbol. I ...
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Oval of Cassini In geometry, a Cassini oval is a quartic plane curve defined as the locus (mathematics), locus of points in the plane (geometry), plane such that the Product (mathematics), product of the distances to two fixed points (Focus (geometry), foci) is ...
*
Cartesian oval In geometry, a Cartesian oval is a plane curve consisting of points that have the same linear combination of distances from two fixed points (foci). These curves are named after French mathematician René Descartes, who used them in optics. Defi ...
*
Ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ...


References

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 c_1 and c_2. This system is very useful in some scientific applications (e.g. calculating the electric field ...
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