  We consider the problem of reconstructing a compact 3-manifold (with
boundary) embedded in $\mathbb{R}^3$ from its cross-sections $\mathcal S$ with
a given set of cutting planes $\mathcal P$ having arbitrary orientations. Using
the obvious fact that a point $x \in \mathcal P$ belongs to the original object
if and only if it belongs to $\mathcal S$, we follow a very natural
reconstruction strategy: we say that a point $x \in \mathbb{R}^3$ belongs to
the reconstructed object if (at least one of) its nearest point(s) in $\mathcal
P$ belongs to $\mathcal S$. This coincides with the algorithm presented by Liu
et al. in \cite{LB+08}. In the present paper, we prove that under appropriate
sampling conditions, the output of this algorithm preserves the homotopy type
of the original object. Using the homotopy equivalence, we also show that the
reconstructed object is homeomorphic (and isotopic) to the original object.
This is the first time that 3-dimensional shape reconstruction from
cross-sections comes with theoretical guarantees.
