# Computation of Circular Area and Spherical Volume Invariants via Boundary Integrals

## TL;DR

This paper presents a method to compute circular area and spherical volume invariants of surfaces using boundary integrals, enabling efficient analysis of 3D shapes without ambient space discretization.

## Contribution

It introduces a boundary integral approach for computing surface invariants, extending to higher dimensions and applicable to triangulated meshes, with potential applications in anthropology.

## Key findings

- Provides analytical formulas for invariants on triangulated meshes
- Enables computation without ambient space discretization
- Applicable to feature detection on 3D surfaces

## Abstract

We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume integrals as line and surface integrals, respectively, against particular kernels; our results also extend to higher dimensional hypersurfaces. The resulting surface integrals are computable analytically on a triangulated mesh. This gives a simple computational algorithm for computing the spherical volume invariant for triangulated surfaces that does not involve discretizing the ambient space. We discuss potential applications to feature detection on broken bone fragments of interest in anthropology.