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20
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 54 (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?
Unfolding Some Classes of Orthogonal Polyhedra
, 1998
"... In this paper, we study unfoldings of orthogonal polyhedra. More precisely, we define two special classes of orthogonal polyhedra, orthostacks and orthotubes, and show how to generate unfoldings by cutting faces, such that the resulting surfaces can be flattened into a single connected polygon. ..."
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Cited by 35 (14 self)
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In this paper, we study unfoldings of orthogonal polyhedra. More precisely, we define two special classes of orthogonal polyhedra, orthostacks and orthotubes, and show how to generate unfoldings by cutting faces, such that the resulting surfaces can be flattened into a single connected polygon.
A motion planning approach to folding: From paper craft to protein folding
 In Proc. IEEE Int. Conf. Robot. Autom. (ICRA
, 2001
"... In this paper, we present a framework for studying folding problems from a motion planning perspective. In particular, all folding objects are modeled as treelike multilink articulated `robots', where fold positions correspond to joints and areas that cannot fold correspond to links. This formulat ..."
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Cited by 29 (9 self)
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In this paper, we present a framework for studying folding problems from a motion planning perspective. In particular, all folding objects are modeled as treelike multilink articulated `robots', where fold positions correspond to joints and areas that cannot fold correspond to links. This formulation allows us to apply recent techniques developed in the robotics motion planning community for articulated objects with many degrees of freedom (many links) to folding problems. An important bene t of this approach is that it not only allows us to study foldability questions, such as, can one object be folded (or unfolded) into another object, but also enables us to study the dynamic folding process itself. The framework proposed here has application in traditional motion planning areas such as automation, teaching through demonstration, animation, and most importantly, presents a di erent approach to the most profound problem in computational biology: protein struction prediction. Indeed, our preliminary experimental results with traditional paper crafts (e.g., box folding) and a relatively small protein are quite promising.
Ununfoldable polyhedra with convex faces
 COMPUT. GEOM. THEORY APPL
, 2002
"... Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex fa ..."
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Cited by 26 (11 self)
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Unfolding a convex polyhedron into a simple planar polygon is a wellstudied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that “open” polyhedra with triangular faces may not be unfoldable no matter how they are cut.
When Can a Polygon Fold to a Polytope?
 Dept. Comput. Sci., Smith College
, 1996
"... We show that the decision question posed in the title can be answered with an algorithm of time and space complexity O(n 2 ), for a polygon of n vertices. We use a theorem of Aleksandrov that says that if the edges of the polygon can be matched in length so that the resulting complex is homeomo ..."
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Cited by 25 (6 self)
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We show that the decision question posed in the title can be answered with an algorithm of time and space complexity O(n 2 ), for a polygon of n vertices. We use a theorem of Aleksandrov that says that if the edges of the polygon can be matched in length so that the resulting complex is homeomorphic to a sphere, and such that the "complete angle" at each vertex is no more than 2, then the implied folding corresponds to a unique convex polytope. We check the Aleksandrov conditions via dynamic programming. The algorithm has been implemented and tested. 1 Introduction The polygon shown in Fig. 1a cannot fold edgetoedge to a convex polytope despite being composed of six squares that one might think could fold to a cube. The familiar "cross" polygon in Fig. 1b can of course fold to a cube. The aim of this paper is to provide an algorithm that can decide whether a polygon can fold to a polytope. We take a polygon P to be a collection of vertices (v 0 ; v 1 ; :::; v n\Gamma1 ) in...
Ununfoldable Polyhedra
, 1999
"... A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can inde ..."
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Cited by 16 (9 self)
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A wellstudied problem is that of unfolding a convex polyhedron into a simple planar polygon. In this paper, we study the limits of unfoldability. We give an example of a polyhedron with convex faces that cannot be unfolded by cutting along its edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that "open" polyhedra with convex faces may not be unfoldable no matter how they are cut.
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
Nets of Polyhedra
, 1997
"... In 1525, the painter Albrecht Dürer introduced the notion of a net of a polytope, and published nets for some of the Platonian and Archimedian polyhedra, along with directions about how to construct them. An unfolding of a 3dimensional polytope P is obtained by cutting the boundary of P along a co ..."
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Cited by 7 (0 self)
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In 1525, the painter Albrecht Dürer introduced the notion of a net of a polytope, and published nets for some of the Platonian and Archimedian polyhedra, along with directions about how to construct them. An unfolding of a 3dimensional polytope P is obtained by cutting the boundary of P along a collection of edges that spans the vertex set of P and then flattening the remaining set to a polygon in the plane. An unfolding is a net if it does not overlap itself. Conversely, a simple connected plane polygon with specific folding lines is a net, if it is possible to fold it into (the boundary of) a polytope. We consider the question whether every 3dimensional polytope has a net. Although the problem is intuitive and easy to state, and there are nets known for all regular and uniform polytopes, in general it is still unsolved. After giving an overview of related questions and conjectures about the nature or existence of nets for 3polytopes, we present an account of our experiments wit...