- #1

vanesch

Staff Emeritus

Science Advisor

Gold Member

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## Main Question or Discussion Point

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.

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.