Efficient Conformal Parameterization of Multiply-Connected Surfaces Using Quasi-Conformal Theory

• Published in: Journal of scientific computing Vol. 87; no. 3; p. 70
• Main Author: Choi, Gary P. T.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Conformal mapping; Differential geometry; Mapping; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical functions; Mathematics; Mathematics and Statistics; Methods; Parameterization; Theoretical; Surface parameterization; 30C20; Conformal mapping; Quasi-conformal theory; 52C26; Annulus; Poly-annulus; 68U05; Multiply-connected surfaces; 65D18
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-021-01479-y

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.