Reconstructing convex polygons and convex polyhedra from edge and face counts in orthogonal projections

TLDR: 3D reconstructing convex polygons and convex polyhedra given the number of visible edges and visible faces in some orthogonal projections is studied and it is shown that the problem becomes NP-hard when the directions are covered by three or more planes.

Abstract: We study the problem of reconstructing convex polygons and convex polyhedra given the number of visible edges and visible faces in some orthogonal projections. In 2D, we find necessary and sufficient conditions for the existence of a feasible polygon of size N and give an algorithm to construct one, if it exists. When N is not known, we give an algorithm to find the maximum and minimum sizes of a feasible polygon. In 3D, when the directions are covered by a single plane we show that a feasible polyhedron can be constructed from a feasible polygon. We also give an algorithm to construct a feasible polyhedron when the directions are covered by two planes. Finally, we show that the problem becomes NP-hard when the directions are covered by three or more planes.