# Conformal map for regular polygon in circle.

Staff Emeritus
Gold Member
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.

Staff Emeritus
Gold Member
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.