# A Physics-Based Estimation of Mean Curvature Normal Vector for Triangulated Surfaces

## TL;DR

This paper introduces a physics-based method for estimating the mean curvature normal vector on triangulated surface meshes, linking physical principles with discrete differential geometry for improved surface analysis.

## Contribution

It derives a new approximation based on the Young-Laplace equation, showing its equivalence to the discrete Laplace-Beltrami operator and extending applicability to various mesh types.

## Key findings

- Derived a physics-based approximation for mean curvature normal vector.
- Proved equivalence to the discrete Laplace-Beltrami operator.
- Applicable to non-triangular and heterogeneous meshes.

## Abstract

In this note, we derive an approximation for the mean curvature normal vector on vertices of triangulated surface meshes from the Young-Laplace equation and the force balance principle. We then demonstrate that the approximation expression from our physics-based derivation is equivalent to the discrete Laplace-Beltrami operator approach in the literature. This work, in addition to providing an alternative expression to calculate the mean curvature normal vector, can be further extended to other mesh structures, including non-triangular and heterogeneous meshes.