  We explore the geometric implications of introducing a spectral cut-off on
Riemannian manifolds. This is naturally phrased in the framework of
non-commutative geometry, where we work with spectral triples that are
\emph{truncated} by spectral projections of Dirac-type operators. We prove that
the underlying Riemannian manifold is the Gromov-Hausdorff limit of the metric
spaces we associate to its truncations. This leads us to propose a
computational algorithm that allows us to recover these metric spaces from the
finite-dimensional truncated spectral data. We subsequently develop a technique
for embedding the resulting metric graphs in Euclidean space to asymptotically
recover an isometric embedding of the limit. We test these algorithms on the
truncated sphere and a recently investigated perturbation thereof.
