Discrete Approximation of Surface Curvature

Title: Discrete Approximation of Surface Curvature

Abstract: This research explores the approximation of surface curvature using discrete, metric spaces. The study focuses on developing metric equivalents of Gaussian curvature for triangulated surfaces, allowing for a more natural and accurate representation of curvature in finite spaces. The research proposes two modalities for computing Gaussian curvature, based on embedding curvature and Wald curvature. The study demonstrates the equivalence between Wald and Gauss curvatures for smooth surfaces, providing a legitimate discrete candidate for approximating Gaussian curvature on triangulated surfaces. The research also presents formulas for computing Wald curvature, contributing to the development of a more accurate and computationally efficient method for evaluating the intrinsic properties of triangulated surfaces.

Main Research Question: Can we develop a metric-based approach to approximate the Gaussian curvature of triangulated surfaces, leading to more accurate and computationally efficient methods for evaluating the intrinsic properties of these surfaces?

Methodology: The research employs a metric approach to study the problem of better approximating surfaces by triangulations. This method considers the triangulations as finite metric spaces and the target smooth surface as their Haussdorff-Gromov limit. By defining relevant elements, constants, and invariants in a more natural, discrete, and metric manner, the study avoids conceptual and computational errors inherent to classical, numerical approaches.

Results: The research presents two modalities for computing Gaussian curvature using embedding and Wald curvature. It demonstrates the equivalence between Wald and Gauss curvatures for smooth surfaces, providing a discrete candidate for approximating Gaussian curvature on triangulated surfaces. The study also presents formulas for computing Wald curvature, contributing to the development of a more accurate and computationally efficient method for evaluating the intrinsic properties of triangulated surfaces.

Implications: The research has significant implications for various fields that rely on the study of triangulated surfaces, such as computer graphics, geometric modeling, and biomedical engineering. By providing a more accurate and computationally efficient method for evaluating the intrinsic properties of triangulated surfaces, the research contributes to the development of better algorithms and models in these fields. Additionally, the study's focus on metric equivalents of Gaussian curvature offers a new perspective on the approximation of continuous surfaces using discrete, metric spaces.

Link to Article: https://arxiv.org/abs/0401023v1 Authors: arXiv ID: 0401023v1