  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.
