  The principles of mesh equidistribution and alignment play a fundamental role
in the design of adaptive methods, and a metric tensor M and mesh metric are
useful theoretical tools for understanding a methods level of mesh alignment,
or anisotropy. We consider a mesh redistribution method based on the
Monge-Ampere equation, which combines equidistribution of a given scalar
density function with optimal transport. It does not involve explicit use of a
metric tensor M, although such a tensor must exist for the method, and an
interesting question to ask is whether or not the alignment produced by the
metric gives an anisotropic mesh. For model problems with a linear feature and
with a radially symmetric feature, we derive the exact form of the metric M,
which involves expressions for its eigenvalues and eigenvectors. The
eigenvectors are shown to be orthogonal and tangential to the feature, and the
ratio of the eigenvalues (corresponding to the level of anisotropy) is shown to
depend, both locally and globally, on the value of the density function and the
amount of curvature. We thereby demonstrate how the optimal transport method
produces an anisotropic mesh along a given feature while equidistributing a
suitably chosen scalar density function. Numerical results are given to verify
these results and to demonstrate how the analysis is useful for problems
involving more complex features, including for a non-trivial time dependant
nonlinear PDE which evolves narrow and curved reaction fronts.
