image of Bipolar


2D Coordinate Systems


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).

Object definitions


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 of Bipolar
<math> <mrow> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow> </math>

Cite this as:

2D Coordinate Systems: Bipolar from Differential Geometry Library.

Terms of use:

You can use the contents of DG Library pages for education, research, professional needs and enjoyment.
If you use this library, please, cite DG Library as the source of the data.

Pages of DG Library may not be copied, mirrored, redistributed, printed, or reproduced in bulk without permission. Usage for any commercial purposes without permission is prohibited.

DG Library is database of over 580 objects for differential geometry and its applications. Read more...

  • Share via email
  • Share via Twitter
  • Share via Facebook