Computing boundary extensions of conformal maps

TL;DR

This paper demonstrates that boundary extensions of conformal maps can be effectively approximated using high-quality approximations of the maps themselves and local connectivity data of the boundary.

Contribution

It introduces a method to compute boundary extensions of conformal maps from approximate maps and boundary connectivity information.

Findings

Boundary extensions can be approximated arbitrarily well.

Good approximations of the conformal map and boundary connectivity suffice.

The approach enables practical computation of boundary extensions.

Abstract

Let $\phi$ be a conformal map of the unit disk onto a domain $D$, and suppose $\phi$ has a boundary extension. We show that arbitrarily good approximations of the boundary extension of $\phi$ can be computed from sufficiently good approximations of $\phi$ and sufficient local connectivity information for the boundary of $D$.