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13
Folding and Unfolding in Computational Geometry
"... 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 ..."
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Cited by 53 (4 self)
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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?
SingleStrip Triangulation of Manifolds with Arbitrary Topology
, 2004
"... Triangle strips have been widely used for efficient rendering. It is NPcomplete 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 cre ..."
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Cited by 16 (5 self)
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Triangle strips have been widely used for efficient rendering. It is NPcomplete 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 singlestrip 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.
Grid vertexunfolding orthostacks
 International Journal of Computational Geometry and Applications
"... Communicated by Godfried Toussaint Biedl et al. 1 presented an algorithm for unfolding orthostacks into one piece without overlap by using arbitrary cuts along the surface. They conjectured that orthostacks could be unfolded using cuts that lie in a plane orthogonal to a coordinate axis and containi ..."
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Cited by 8 (1 self)
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Communicated by Godfried Toussaint Biedl et al. 1 presented an algorithm for unfolding orthostacks into one piece without overlap by using arbitrary cuts along the surface. They conjectured that orthostacks could be unfolded using cuts that lie in a plane orthogonal to a coordinate axis and containing a vertex of the orthostack. We prove the existence of a vertex unfolding using only such cuts.
When Can a Net Fold to a Polyhedron?
 In Proceedings of the 11th Canadian Conference on Computational Geometry
, 1999
"... 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 ..."
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Cited by 8 (1 self)
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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 NPcomplete, 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.
Reconfigurations of polygonal structures
, 2005
"... 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 polygona ..."
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Cited by 8 (1 self)
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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 noncrossing straight edges joining them. A polyhedral surface is a threedimensional 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 welldefined way of transforming one instance of a structure 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 unitlength edges and angles 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
Epsilonunfolding orthogonal polyhedra
 Graphs and Combinatorics
"... 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 o ..."
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Cited by 7 (2 self)
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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 ǫ = 1/2 Ω(n). 1
Grid vertexunfolding orthogonal polyhedra
 In Proc. 23rd Sympos. Theoret. Aspects Comput. Sci., Lecture Notes Comput. Sci
, 2006
"... An edgeunfolding 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 t ..."
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Cited by 4 (1 self)
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An edgeunfolding 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 polyhedron of genus zero has a grid vertexunfolding. (There are orthogonal polyhedra that cannot be vertexunfolded, so some type of “gridding ” of the faces is necessary.) For any orthogonal polyhedron P with n vertices, we describe an algorithm that vertexunfolds P in O(n 2) time. Enroute to explaining this algorithm, we present two simpler vertexunfolding algorithms, one for “banded ” objects, and one that requires a 3 ×1 refinement of the vertex grid. 1
Hinged dissections exist
"... We prove that any finite collection of polygons of equal area has a common hinged dissection, that is, a chain of polygons hinged at vertices that can be folded in the plane continuously without selfintersection to form any polygon in the collection. This result settles the open problem about the e ..."
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Cited by 3 (1 self)
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We prove that any finite collection of polygons of equal area has a common hinged dissection, that is, a chain of polygons hinged at vertices that can be folded in the plane continuously without selfintersection to form any polygon in the collection. This result settles the open problem about the existence of hinged dissections between pairs of polygons that goes back implicitly to 1864 and has been studied extensively in the past ten years. Our result generalizes and indeed builds upon the result from 1814 that polygons have common dissections (without hinges). We also extend our result to edgehinged dissections of solid 3D polyhedra that have a common (unhinged) dissection, as determined by Dehn’s 1900 solution to Hilbert’s Third Problem. Our proofs are constructive, giving explicit algorithms in all cases. For a constant number of planar polygons, both the number of pieces and running time required by our construction are pseudopolynomial. This bound is the best possible even for unhinged dissections. Hinged dissections have possible applications to reconfigurable robotics, programmable matter, and nanomanufacturing.