Distortion estimates for barycentric coordinates on Riemannian simplices

TL;DR

This paper introduces a method to define and analyze barycentric coordinates on Riemannian manifolds using Karcher's center of mass, providing estimates that facilitate finite element approximations on curved spaces.

Contribution

It develops a new approach to construct Riemannian simplices via Karcher's center of mass and derives derivative estimates for the coordinate maps, enabling metric comparisons and convergence analysis.

Findings

The coordinate map is smooth and injective for points in general position.

The metric difference between Riemannian and Euclidean metrics shrinks quadratically with edge length.

These estimates support convergence results for finite element methods on manifolds.

Abstract

We define barycentric coordinates on a Riemannian manifold using Karcher's center of mass technique applied to point masses for n+1 sufficiently close points, determining an n-dimensional Riemannian simplex defined as a "Karcher simplex." Specifically, a set of weights is mapped to the Riemannian center of mass for the corresponding point measures on the manifold with the given weights. If the points lie sufficiently close and in general position, this map is smooth and injective, giving a coordinate chart. We are then able to compute first and second derivative estimates of the coordinate chart. These estimates allow us to compare the Riemannian metric with the Euclidean metric induced on a simplex with edge lengths determined by the distances between the points. We show that these metrics differ by an error that shrinks quadratically with the maximum edge length. With such estimates, one can deduce convergence results for finite element approximations of problems on Riemannian manifolds.