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.