# Conformal mapping

1. Oct 8, 2014

### JulieK

Is there a conformal mapping that transforms regular polygons (e.g. triangle and square) to circle?

2. Oct 8, 2014

### Staff: Mentor

3. Oct 11, 2014

### homeomorphic

4. Oct 11, 2014

### lavinia

As homeomorphic wrote, the Riemann Mapping Theorem solves this problem.

A couple of points for clarification

The theorem applies to non-empty simply connected open domains in the complex plane other than the entire plane. The interior of a polygon and a circle are both simply connected. Thus there is a conformal mapping from the interior of the polygon onto the interior of the circle. The bounding polygon is mapped onto the bounding circle but the map is obviously not conformal on these edges and can never be ( as you suspected).

The boundary of a simply connected domain can be complicated and need not be piece wise smooth. Nevertheless its interior can be mapped conformally onto the interior of a circle.

Last edited: Oct 11, 2014
5. Oct 13, 2014

### JulieK

Thank you all!

6. Oct 14, 2014

### jasonRF

The Schwarz-Christoffel transformation will map a polygon to the upper half plane, and a Mobius transformation can map a half-plane to the unit circle.