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.

保形贴图是复杂分析和微分几何学中的经典主题，在近几十年来，随着科学和工程的各种应用，保形贴图已经成为表面参数化领域的一个重要课题。但是，大多数现有的共形参数化算法仅关注简单连接的曲面，而不能直接应用于带有孔的曲面。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曲面的保形参数化。我们首先开发了一种有效的方法，用于将带有一个孔的开放表面共形化参数化到平面上的环形空间。基于该方法，我们然后开发一种有效的方法，用于共形地参数化的开放表面与ķ孔到与单位圆盘ķ圆孔。映射的共形性和双射性是通过准保形理论来保证的。数值实验和应用被提出来证明所提出的方法的有效性。