  A continuum description of unstructured meshes in two dimensions, both for
planar and curved surface domains, is proposed. The meshes described are those
which, in the limit of an increasingly finer mesh (smaller cells), and away
from irregular vertices, have ideally-shaped cells (squares or equilateral
triangles), and can therefore be completely described by two local properties:
local cell size and local edge directions. The connection between the two
properties is derived by defining a Riemannian manifold whose geodesics trace
the edges of the mesh. A function $\phi$, proportional to the logarithm of the
cell size, is shown to obey the Poisson equation, with localized charges
corresponding to irregular vertices. The problem of finding a suitable manifold
for a given domain is thus shown to exactly reduce to an Inverse Poisson
problem on $\phi$, of finding a distribution of localized charges adhering to
the conditions derived for boundary alignment. Possible applications to mesh
generation are discussed.
