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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?
When can you fold a map
 In Proceedings of the 7th International Workshop on Algorithms and Data Structures
, 2001
"... Abstract. We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest onelayer simple fold rotates a portio ..."
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Cited by 8 (4 self)
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Abstract. We explore the following problem: given a collection of creases on a piece of paper, each assigned a folding direction of mountain or valley, is there a flat folding by a sequence of simple folds? There are several models of simple folds; the simplest onelayer simple fold rotates a portion of paper about a crease in the paper by ¦ � Æ. We first consider the analogous questions in one dimension lower—bending a segment into a flat object—which lead to interesting problems on strings. We develop efficient algorithms for the recognition of simply foldable 1D crease patterns, and reconstruction of a sequence of simple folds. Indeed, we prove that a 1D crease pattern is flatfoldable by any means precisely if it is by a sequence of onelayer simple folds. Next we explore simple foldability in two dimensions, and find a surprising contrast: “map ” folding and variants are polynomial, but slight generalizations are NPcomplete. Specifically, we develop a lineartime algorithm for deciding foldability of an orthogonal crease pattern on a rectangular piece of paper, and prove that it is (weakly) NPcomplete to decide foldability of (1) an orthogonal crease pattern on a orthogonal piece of paper, (2) a crease pattern of axisparallel and diagonal (45degree) creases on a square piece of paper, and (3) crease patterns without a mountain/valley assignment. 1
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
Survey on Modelbased Manipulation Planning of Deformable Objects
"... A systematic overview on the subject of modelbased manipulation planning of deformable objects is presented. Existing modelling techniques of volumetric, planar and linear deformable objects are described, emphasizing the different types of deformation. Planning strategies are categorized according ..."
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Cited by 1 (0 self)
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A systematic overview on the subject of modelbased manipulation planning of deformable objects is presented. Existing modelling techniques of volumetric, planar and linear deformable objects are described, emphasizing the different types of deformation. Planning strategies are categorized according to the type of manipulation goal: path planning, folding/unfolding, topology modifications and assembly. Most current contributions fit naturally into these categories, and thus the presented algorithms constitute an adequate basis for future developments. Key words: robotic manipulation, manipulation planning, deformable objects, deformable models 1.
Robot SelfAssembly by Folding: A Printed Inchworm Robot
"... Abstract — Printing and folding are fast and inexpensive methods for prototyping complex machines. Selfassembly of the folding step would expand the possibilities of this method to include applications where external manipulation is costly, such as microassembly, mass production, and space applica ..."
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Abstract — Printing and folding are fast and inexpensive methods for prototyping complex machines. Selfassembly of the folding step would expand the possibilities of this method to include applications where external manipulation is costly, such as microassembly, mass production, and space applications. This paper presents a method for selffolding of printed robots from twodimensional materials based on shape memory polymers actuated by joule heating using embedded circuits. This method was shown to be capable of sequential folding, anglecontrolled folds, slotandtab assembly, and mountain and valley folds. An inchworm robot was designed to demonstrate the merits of this technique. Upon the application of sufficient current, the robot was able to fold into its functional form with fold angle deviations within six degrees. This printed robot demonstrated locomotion at a speed of two millimeters per second.