# Weak and approximate curvatures of a measure: a varifold perspective

## TL;DR

This paper introduces a new framework for defining and computing weak and approximate curvature tensors for varifolds, including point clouds, with proven convergence and demonstrated numerical effectiveness.

## Contribution

It extends Hutchinson's generalized second fundamental form to all varifolds via regularization, enabling curvature analysis of diverse datasets like point clouds.

## Key findings

- Explicit formulas for curvature tensors
- Structural properties and convergence proofs
- Numerical tests confirm effectiveness

## Abstract

By revisiting the notion of generalized second fundamental form originally introduced by Hutchinson for a special class of integral varifolds, we define a weak curvature tensor that is particularly well-suited for being extended to general varifolds of any dimension and codimension through regularization. The resulting approximate second fundamental forms are defined not only for piecewise-smooth surfaces, but also for datasets of very general type (like, e.g., point clouds). We obtain explicitly computable formulas for both weak and approximate curvature tensors, we exhibit structural properties and prove convergence results, and lastly we provide some numerical tests on point clouds that confirm the generality and effectiveness of our approach.