Geodesic distance approximation using a surface finite element method for the $p$-Laplacian

TL;DR

This paper introduces a surface finite element method using the $p$-Laplacian with large $p$-values to accurately approximate geodesic distances on surfaces, demonstrating convergence, robustness, and advantages over existing PDE-based methods.

Contribution

It presents a novel finite element approach employing the $p$-Laplacian for geodesic distance approximation, with numerical validation and comparison to established methods.

Findings

Numerical convergence to true geodesic distances.

Method adheres to the triangle inequality.

Robustness against geometric noise.

Abstract

We use the $p$-Laplacian with large $p$-values in order to approximate geodesic distances to features on surfaces. This differs from Fayolle and Belyaev's (2018) [1] computational results using the $p$-Laplacian for the distance-to-surface problem. Our approach appears to offer some distinct advantages over other popular PDE-based distance function approximation methods. We employ a surface finite element scheme and demonstrate numerical convergence to the true geodesic distance functions. We check that our numerical results adhere to the triangle inequality and examine robustness against geometric noise such as vertex perturbations. We also present comparisons of our method with the heat method from Crane et al. [2] and the classical polyhedral method from Mitchell et al. [3].