# Barycentric coordinate neighbourhoods in Riemannian manifolds

arXiv: https://arxiv.org/abs/1606.01585 · HTML: https://mlsanity.com/paper/1606.01585/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2016.

## Built on by (most cited first, 2 of 2)

- [Five Degree-of-Freedom Property Interpolation of Arbitrary Grain Boundaries via Voronoi Fundamental Zone Octonion Framework](https://mlsanity.com/paper/2104.06575/) (2021)
- [Fr'echet Regression on the Bures-Wasserstein Manifold](https://mlsanity.com/paper/2604.03566/) (2026)

## Abstract (arXiv)

We quantify conditions that ensure that a signed measure on a Riemannian manifold has a well defined centre of mass. We then use this result to quantify the extent of a neighbourhood on which the Riemannian barycentric coordinates of a set of n+1 points on an n-manifold provide a true coordinate chart, i.e., the barycentric coordinates provide a diffeomorphism between a neighbourhood of a Euclidean simplex, and a neighbourhood containing the points on the manifold.