The shadow of an abstract simplicial complex \(\mathcal {K}\) with vertices in \(\mathbb {R}^N\) is a subset of \(\mathbb {R}^N\) defined as the union of the convex hulls of simplices of \(\mathcal {K}\) . The Vietoris–Rips complex of a metric space \((\mathcal {S},d)\) at scale \(\beta \) is an abstract simplicial complex whose each k-simplex corresponds to \((k+1)\) points of \(\mathcal {S}\) within diameter \(\beta \) . In case \(\mathcal {S}\subset \mathbb {R}^2\) and \(d(a,b)=\Vert a-b\Vert \) the standard Euclidean metric, the natural shadow projection of the Vietoris–Rips complex is already proved by Chambers et al. to induce isomorphisms on \(\pi _0\) and \(\pi _1\) . We extend the result beyond the standard Euclidean distance on \(\mathcal {S}\subset \mathbb {R}^N\) to a family of path-based metrics, \(d^\varepsilon _{\mathcal {S}}\) . From the pairwise Euclidean distances of points in \(\mathcal {S}\) , we introduce a family (parametrized by \(\varepsilon \) ) of path-based Vietoris–Rips complexes \(\mathcal {R}^\varepsilon _\beta (\mathcal {S})\) for a scale \(\beta >0\) . If \(\mathcal {S}\subset \mathbb {R}^2\) is Hausdorff-close to a planar Euclidean graph \(\mathcal {G}\) , we provide quantitative bounds on scales \(\beta ,\varepsilon \) for the shadow projection map of the Vietoris–Rips complex of \((\mathcal {S},d^\varepsilon _\mathcal {S})\) at scale \(\beta \) to induce \(\pi _1\) -isomorphism. This paper first studies the homotopy-type recovery of \(\mathcal {G}\subset \mathbb {R}^N\) using the abstract Vietoris–Rips complex of a Hausdorff-close sample \(\mathcal {S}\) under the \(d^\varepsilon _\mathcal {S}\) metric. Then, our result on the \(\pi _1\) -isomorphism induced by the shadow projection lends itself to providing also a geometrically close embedding for the reconstruction. Based on the length of the shortest loop and large-scale distortion of the embedding of \(\mathcal {G}\) , we quantify the choice of a suitable sample density \(\varepsilon \) and a scale \(\beta \) at which the shadow of \(\mathcal {R}^\varepsilon _\beta (\mathcal {S})\) is homotopy-equivalent and Hausdorff-close to \(\mathcal {G}\) .
