Three open problems on folding/unfolding are discussed: (1) Can every convex polyhedron be cut along edges and unfolded at to a single nonoverlapping piece? (2) Given gluing instructions for a polygon, construct the unique 3D convex polyhedron to which itfolds. (3) Can every planar polygonal chain be straightened?

Triangle strips have been widely used for efficient rendering. It is NP-complete to test whether a given triangulated model can be represented as a single triangle strip, so many heuristics have been proposed to partition models into few long strips. In this paper, we present a new algorithm for creating a single triangle loop or strip from a triangulated model. Our method applies a dual graph matching algorithm to partition the mesh into cycles, and then merges pairs of cycles by splitting adjacent triangles when necessary. New vertices are introduced at midpoints of edges and the new triangles thus formed are coplanar with their parent triangles, hence the visual fidelity of the geometry is not changed. We prove that the increase in the number of triangles due to this splitting is 50 % in the worst case, however for all models we tested the increase was less than 2%. We also prove tight bounds on the number of triangles needed for a single-strip representation of a model with holes on its boundary. Our strips can be used not only for efficient rendering, but also for other applications including the generation of space filling curves on a manifold of any arbitrary topology.

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.

This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of non-crossing straight edges joining them. A polyhedral surface is a three-dimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a well-defined way of transforming one instance of a struc-ture to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unit-length edges and an-gles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal

Abstract. An unfolding of a polyhedron is produced by cutting the surface and flattening to a single, connected, planar piece without overlap (except possibly at boundary points). It is a long unsolved problem to determine whether every polyhedron may be unfolded. Here we prove, via an algorithm, that every orthogonal polyhedron (one whose faces meet at right angles) of genus zero may be unfolded. Our cuts are not necessarily along edges of the polyhedron, but they are always parallel to polyhedron edges. For a polyhedron of n vertices, portions of the unfolding will be rectangular strips which, in the worst case, may need to be as thin as ε = 1/2 Ω(n).

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An edge-unfolding of a polyhedron is produced by cutting along edges and flattening the faces to a net, a connected planar piece with no overlaps. A grid unfolding allows additional cuts along grid edges induced by coordinate planes passing through every vertex. A vertexunfolding permits faces in the net to be connected at single vertices, not necessarily along edges. We show that any orthogonal polyhedra of genus zero has a grid vertex-unfolding. (There are orthogonal polyhedra that cannot be vertex-unfolded, so some type of “gridding ” of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertex-unfolds P in O(n 2) time. Enroute to explaining this algorithm, we present a simpler vertex-unfolding algorithm that requires a 3 × 1 refinement of the vertex grid.

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The Unfolding Problem can be succinctly described as “How to peel an orange... no matter what shape the orange is. ” It is the question of how to ‘unwrap ’ a 3D polyhedron, breaking some of its edges or faces so that it can be unfolded into a flat net in the 2D plane. From there the flattened net of faces might be printed out, cut from paper or steel and folded to recreate the virtual model in the real world. In 1525 the artist Albrecht Dürer used the term ‘net ’ to describe a set of polygons linked together edge-to-edge to form the planar unfoldings of some of the platonic solids and their truncations. Dürer used these unfoldings to teach aspiring artists how to construct elemental forms, but today the applications for solutions to the unfolding problem lie in a broad range of fields, from industrial manufacturing and rapid prototyping to sculpture and aeronautics. In the textiles industry, work has already begun in computing digital representations of fabric and trying to flatten those representations to optimize seam and dart placement [MHC05]. There has been similar work in the fields of paper-folding [MS04] and origami [BM04] and even ship and sail manufacturing. Advances in robotics and folding automation [GBKK98] have brought with them a new need for faster, more robust unfolding methods. If a polyhedron can generate a net which is not self-intersecting, solely by breaking a subset of its edges and flattening the join angles of those which remain, then it is called edge-unfoldable or developable. At present, it is strongly believed–but not yet proven–that all convex surfaces are developable. In counterpoint, examples are easily found of non-convex surfaces which are cannot be edge-unfolded, but no robust solution yet exists for testing whether or not a given mesh will prove to be developable.

We construct a polyhedron that is topologically convex (i.e., has the graph of a convex polyhedron) yet has no vertex unfolding: no matter how we cut along the edges and keep faces attached at vertices to form a connected (hinged) surface, the surface necessarily unfolds with overlap.