Reconstructing Convex Polygons and Polyhedra from Edge and Face Counts in Orthogonal Projections

We study the problem of constructing convex polygons and convex polyhedra given the number of visible edges and visible faces from 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 size of a feasible polygon. In 3D, when the directions span 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-complete for three or more planes.