Efficient conformal parameterization of multiply-connected surfaces using quasi-conformal theory

TL;DR

This paper introduces two efficient algorithms for conformally parameterizing multiply-connected surfaces with holes, leveraging quasi-conformal theory to ensure conformality and bijectivity, and demonstrating their effectiveness through numerical experiments.

Contribution

The paper presents novel algorithms for conformal parameterization of multiply-connected surfaces, extending existing methods to handle surfaces with multiple holes using quasi-conformal theory.

Findings

Algorithms effectively parameterize surfaces with holes

Mappings are conformal and bijective

Numerical experiments validate the methods

Abstract

Conformal mapping, a classical topic in complex analysis and differential geometry, has become a subject of great interest in the area of surface parameterization in recent decades with various applications in science and engineering. However, most of the existing conformal parameterization algorithms only focus on simply-connected surfaces and cannot be directly applied to surfaces with holes. In this work, we propose two novel algorithms for computing the conformal parameterization of multiply-connected surfaces. We first develop an efficient method for conformally parameterizing an open surface with one hole to an annulus on the plane. Based on this method, we then develop an efficient method for conformally parameterizing an open surface with $k$ holes onto a unit disk with $k$ circular holes. The conformality and bijectivity of the mappings are ensured by quasi-conformal theory. Numerical experiments and applications are presented to demonstrate the effectiveness of the proposed methods.