Beltrami coefficient and angular distortion of discrete geometric mappings

2026-02-08

AI Summary

This study addresses the lack of a precise theoretical characterization of the relationship between Beltrami coefficients and angular distortion in geometric mappings. It establishes, for the first time, a concise quantitative relationship between the norm of the Beltrami coefficient and the absolute angular distortion of individual elements on discrete triangular meshes, yielding an explicit estimate for the maximal angular distortion. Drawing upon quasiconformal theory and discrete differential geometry, and supported by numerical experiments, the work systematically analyzes the angular distortion behavior of conformal, quasiconformal, and area-preserving mapping algorithms. Experimental validation on diverse biological and engineering surface meshes confirms the theoretical findings, providing both a rigorous theoretical foundation and a practical tool for assessing conformality in geometric mappings.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

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.

Problem

• Beltrami coefficient
• angular distortion
• geometric mappings
• conformality
• quasi-conformal theory

Innovation

• Beltrami coefficient
• angular distortion
• quasi-conformal mapping
• conformality
• discrete geometric mapping