Conformal map for regular polygon in circle.

[vanesch]

Hi All,

I'm looking for the conformal mapping (using complex functions) that maps the unit circle (or the upper half plane) into a REGULAR polygon with n vertices. I know the Schwarz-Christoffel transformation for an ARBITRARY polygon, but that doesn't help me because the expression is way too complex to be integrated (I'm trying to find the mapping for a polygon with 120 vertices). I was hoping that the fact that the polygon is REGULAR would simplify the problem. I used the mapping on the unit circle in the S-C transform because out of the symmetry of the problem, that allowed me (I would guess) to fix the unknown images of the vertices: they should also be on a regular polygon. But nevertheless, I cannot solve the integral beyond n = 4.

 

[vanesch]

Ok, I think I found it. For a regular m-polygon, the mapping between the unit circle (z) and the polygon (w) is given by:

dw/dz = 1/(z^m - 1)^(2/m)

which, according to Mathematica, integrates to:

z (1-z^m)^(2/m) (-1 + z^m)^(-2/m) Hypergeometric2F1[1/m, 2/m, 1+1/m,z^m]

and numerically this does indeed reproduce a polygon...

cheers,
Patrick.

 

[tacman]

Hi there,

Conformal maps are commonly used in mathematics to preserve angles and shapes between two different surfaces. In the case of a regular polygon inscribed in a circle, the conformal map can be constructed using the Schwarz-Christoffel transformation, which is a common technique for mapping polygons onto the upper half plane or unit circle.

However, as you have mentioned, the expression for this transformation can become quite complex, especially for polygons with a large number of vertices. In this case, you may want to consider using a numerical method, such as the finite element method, to approximate the conformal map. This would involve dividing the polygon into smaller, simpler shapes and then using numerical integration techniques to compute the map.

Another approach could be to use a different conformal map that is specifically designed for regular polygons. For example, the Koebe-Andreev-Thurston theorem provides a conformal map for regular polygons with an arbitrary number of vertices. This may be a more suitable option for your problem.

Overall, finding a conformal map for a regular polygon in a circle can be a challenging task, especially for polygons with a large number of vertices. However, with the right techniques and approaches, it is possible to find a solution that accurately maps the polygon onto the circle. I hope this helps and good luck with your research!