When can a net fold to a polyhedron

In this paper, we study the problem of whether a polyhedron can be obtained from a net , i.e., a polygon and a set of creases, by folding along the creases. We consider two cases, depending on whether we are given the dihedral angle at each crease. If these dihedral angles are given the problem can be solved in polynomial time by the simple expedient of performing the folding. If the dihedral angles are not given the problem is NP-complete, at least for orthogonal polyhedra. We then turn to the actual folding process, and show an example of a net with rigid faces that can, in the sense above, be folded to form an orthogonal polyhedron, but only by allowing faces to intersect each other during the folding process. In the existing literature, a few related problems have been studied. Shephard investigated when a convex polyhedron has a convex net [She75]. Lubiw and O'Rourke showed how to test in O(n) time whether an n-vertex polygon (with unknown creases) can be folded into a convex polyhedron [LO96]. Another problem is the reverse of ours: Given a polyhedron, can one obtain a net? This can be done for all convex polyhedra [AO92], as well as for some classes of orthogonal polyhedra [BDD98]. A fundamental open problem in this area is whether for any convex polyhedron there exists a net such that the edges of the net are also edges of the convex polyhedron, a so-called unfolding with edge cuts only. Related to our problem of folding a rigid net is the problem of straightening rigid linkages in 3D, which has been studied in [BDD99]. In fact, our proof that some rigid net cannot be folded without intersecting faces is based on the fact that there exists a linkage in 3D that cannot be straightened without having links intersect [CJ98, BDD99].