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Recently, while reading about the Möbius transformation, I found this picture:

enter image description here

As written on the Wikipedia page, this represents an hyperbolic Möbius transformation. Does this remind you of anything?

Personally, when I look at the blue lines in this picture, I think equipotential lines of a charge dipole:

enter image description here

The red lines in the first picture also seem to correspond to the field lines in the second one. I just learned about the existence of Möbius transformations, so unfortunately I'm not sure what the red and blue lines are supposed to represent exactly.

I know that there is a connection between 2D electrostatics and complex numbers: after all, complex number are somehow similar to (although more general than) 2D vectors.

However, now I'm wondering: what is the exact connection, if it exists, between Möbius transformations and equipotential lines in 2D electrostatics?

valerio
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  • The equipotential lines and field lines of a dipole are not circles, see my answer here : Trajectory of electric field lines. – Frobenius Mar 03 '22 at 00:07
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    The transformation you refer to is not Möbius but a kind of conformal transformation known as Schwarz-Christoffel transformation (the method was first published by Schwarz and Christoffel independently of each other). Generally this transformation opens out the interior of a polygon in the $:z-$ plane to the upper half of the $:w-$plane. In our case of Collinear Source and Sink it's similar to Möbius \begin{equation} z=\ln\dfrac{w-a}{w+a} \tag{a}\label{a} \end{equation} – Frobenius Mar 03 '22 at 00:19
  • Reference : $''$ Conformal Transformations in Electrical Engineering $''$ by William John Gibbs ($\ne$ JOSIAH WILLARD GIBBS), CHAPTER 7, $\S$ Collinear Source and Sink. – Frobenius Mar 03 '22 at 00:31
  • Page 83 , page 84, page 85, page 86. – Frobenius Mar 03 '22 at 00:51
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    Thanks for the replies, @Frobenius. I know that equipotential lines of a dipole are not circles. Still, I find the similarity to be remarkable. It seems to me that your comment on the Schwerz-Christoffel transformation could maybe be expanded into an answer? – valerio Mar 03 '22 at 09:35

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