# Reconstruct polyhedron from sections

There is a convex polyhedron $P \subset \mathbb{R}^{3}$ and there are its planar sections $S_{1}, \ldots, S_{n}$ througth planes $\pi_{1}, \ldots, \pi_{n}$, $S_{i} \subset \pi_{i}$. All these $S_{i}$ are planar polygons.

We don't know neither $P$ nor its sections $S_{1}, \ldots, S_{n}$.

We only know $S^{*}_{1}, \ldots, S^{*}_{n}$ that are noisy measurements of $S_{1}, \ldots, S_{n}$, $S^{*}_{i} \subset \pi_{i}$. It's suggested that for each $i = 1, \ldots, n$ both $S^{*}_{i}$ and $S_{i}$ has the same number of sides and vertices, i. e. they have the same topologycal structure.

Are there any ideas how to reconstruct the convex polyhedron $P^{*}$ from $S^{*}_{1}, \ldots, S^{*}_{n}$ which will be the best approximation of $P$ by these measurements?

• Do we know the $\pi_i$ exactly? – joriki Jul 16 '15 at 10:08
• @joriki, yes, these planes are known. – Ilya Palachev Jul 17 '15 at 10:20
• And if you just take the convex closure of the union of $S_1^*,...,S_n^*$? – san Jul 23 '15 at 4:52
• @san, The problem is that points are noisy and just ~50% of them lie on the convex hull. Other 50% will be ignored. And, for example, if one of points has a big errorm it will cause a lot of points that are close to it to be ignored, since they will not be present in the convex hull. – Ilya Palachev Jul 23 '15 at 8:54