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.

中文翻译：

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使用准共形理论对多重连接表面进行有效的共形参数化

保形映射是复分析和微分几何中的一个经典主题，近几十年来，随着科学和工程中的各种应用，它已成为表面参数化领域的一个非常感兴趣的主题。然而，现有的共形参数化算法大多只关注简单连接的曲面，不能直接应用于带孔的曲面。在这项工作中，我们提出了两种新颖的算法来计算多重连接表面的共形参数化。我们首先开发了一种有效的方法，用于将带一个孔的开放表面共形参数化到平面上的环面。基于这种方法，我们开发了一种有效的方法，将具有 $k$ 孔的开放表面共形参数化到具有 $k$ 圆孔的单位圆盘上。映射的共形和双射性由准共形理论保证。数值实验和应用被提出来证明所提出方法的有效性。