📝 SummarXiv

Abstract

Three-dimensional (3D) mappings are fundamental in various scientific and engineering applications, including computer-aided engineering (CAE), computer graphics, and medical imaging. They are typically represented and stored as three-dimensional coordinates to which each vertex is mapped. With this representation, manipulating 3D mappings while preserving desired properties becomes challenging. In this work, we present a novel geometric representation for 3D bijective mappings, termed 3D quasiconformality (3DQC), which generalizes the concept of Beltrami coefficients from 2D to 3D spaces. This geometric representation facilitates the scientific computation of 3D mapping problems by capturing local geometric properties in 3D mappings. We derive a partial differential equation (PDE) that links the 3DQC to its corresponding mapping. This PDE is discretized into a symmetric positive-definite linear system, which can be efficiently solved using the conjugate gradient method. 3DQC offers a powerful tool for manipulating 3D mappings while maintaining their desired geometric properties. Leveraging 3DQC, we develop numerical algorithms for sparse modeling and numerical interpolation of bijective 3D mappings, facilitating the efficient processing, storage, and manipulation of complex 3D mappings while ensuring bijectivity. Extensive numerical experiments validate the effectiveness and robustness of our proposed methods.