  We consider the topological and geometric reconstruction of a geodesic
subspace of $\mathbb{R}^N$ both from the \v{C}ech and Vietoris-Rips filtrations
on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique
leverages the intrinsic length metric induced by the geodesics on the subspace.
We consider the distortion and convexity radius as our sampling parameters for
a successful reconstruction. For a geodesic subspace with finite distortion and
positive convexity radius, we guarantee a correct computation of its homotopy
and homology groups from the sample. For geodesic subspaces of $\mathbb{R}^2$,
we also devise an algorithm to output a homotopy equivalent geometric complex
that has a very small Hausdorff distance to the unknown shape of interest.
