image of Bipolar

Bipolar

2D Coordinate Systems

Description

Bipolar coordinates are a two-dimensional orthogonal coordinate system.
Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates (see Bipolar cylindrical, Bispherical, Toroidal).

References

Moon & Spencer
Murray R. Spiegel

Object definitions

Mapping

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Mapping of Bipolar
\left\{x\to \frac{a \sinh (v)}{\cosh (v)-\cos (u)},y\to \frac{a \sin (u)}{\cosh (v)-\cos (u)}\right\}
<math> <mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mfrac> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mrow> <mi>sinh</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mi>cosh</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mfrac> <mrow> <mi>a</mi> <mo>&#8290;</mo> <mrow> <mi>sin</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mi>cosh</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </math>
{x -> (a*Sinh[v])/(-Cos[u] + Cosh[v]), y -> (a*Sin[u])/(-Cos[u] + Cosh[v])}
[x = a*sinh(v)/(-cos(u)+cosh(v)), y = a*sin(u)/(-cos(u)+cosh(v))]

Constants

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Constants of Bipolar
\{a\}
<math> <mrow> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow> </math>
{a}
[a]

Cite this as:

2D Coordinate Systems: Bipolar from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/2DCoordinateSystems/Bipolar.php

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