Over the past several decades, geometric mapping methods have been extensively developed and utilized for many practical problems in science and engineering. To assess the quality of geometric mappings, one common consideration is their conformality. In particular, it is well-known that conformal mappings preserve angles and hence the local geometry, which is beneficial in many applications. Therefore, many existing works have focused on the angular distortion as a measure of the conformality of mappings. More recently, quasi-conformal theory has attracted increasing attention in the development of geometric mapping methods, in which the Beltrami coefficient has also been considered as a representation of the conformal distortion. However, the precise connection between these two concepts has not been analyzed. In this work, we study the connection between the two concepts and establish a series of theoretical results. In particular, we discover a simple relationship between the norm of the Beltrami coefficient of a mapping and the absolute angular distortion of triangle elements under the mapping. We can further estimate the maximal angular distortion using a simple formula in terms of the Beltrami coefficient. We verify the developed theoretical results and estimates using numerical experiments on multiple geometric mapping methods, covering conformal mapping, quasi-conformal mapping, and area-preserving mapping algorithms, for a variety of surface meshes in biology and engineering. Altogether, by establishing the theoretical foundation for the relationship between the angular distortion and Beltrami coefficient, our work opens up new avenues for the quantification and analysis of surface mapping algorithms.

翻译：过去几十年间，几何映射方法已广泛应用于科学与工程领域的诸多实际问题。评估几何映射质量时，共形性是一个常见考量指标。特别地，众所周知共形映射能保持角度不变，从而保留局部几何结构，这在众多应用中具有重要价值。因此，现有研究多聚焦于将角度畸变作为映射共形性的度量。近年来，拟共形理论在几何映射方法的发展中日益受到关注，其中贝尔特拉米系数被视为共形畸变的表征。然而，这两个概念间的精确联系尚未被系统分析。本研究探讨了两者之间的关联，并建立了一系列理论结果。具体而言，我们发现了映射的贝尔特拉米系数范数与三角形单元在该映射下的绝对角度畸变之间的简单关系，进而可通过基于贝尔特拉米系数的简洁公式估计最大角度畸变。我们通过涵盖共形映射、拟共形映射及保面积映射算法的多种几何映射方法，对生物与工程领域各类曲面网格进行数值实验，验证了所建立的理论结果与估计。综上，通过构建角度畸变与贝尔特拉米系数关系的理论基础，本研究为曲面映射算法的量化分析开辟了新途径。