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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?
Folding and Unfolding
 in Computational Geometry. 2004. Monograph in preparation
, 2001
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, ..."
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Cited by 16 (4 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, and for that wonderful experience I owe many thanks. I had two excellent advisors, Anna Lubiw and Ian Munro. I started working with Anna after I took her two classes on algorithms and computational geometry during my Master’s, which got me excited about both these areas, and even caused me to switch entire fields of computer science, from distributed systems to theory and algorithms. Anna introduced me to Ian when some of our problems in computational geometry turned out to have large data structural components, and my work with Ian blossomed from there. The sets of problems I worked on with Anna and Ian diverged, and both remain my primary interests. Anna and Ian have had a profound influence throughout my academic career. At the most
Convexifying Polygons with Simple Projections
, 2000
"... It is known that not all polygons in 3D can be convexified when crossing edges are not permitted during any motion. In this paper we prove that if a 3D polygon admits a noncrossing orthogonal projection onto some plane, then the 3D polygon can be convexified. If an algorithm to convexify the planar ..."
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Cited by 8 (2 self)
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It is known that not all polygons in 3D can be convexified when crossing edges are not permitted during any motion. In this paper we prove that if a 3D polygon admits a noncrossing orthogonal projection onto some plane, then the 3D polygon can be convexified. If an algorithm to convexify the planar projection exists and runs in time P , then our algorithm to convexify the 3D polygon runs in O(n + P ) time. By published results, this implies algorithms for any polygon with a convex, monotonic, or starshaped projection.
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
Refolding Planar Polygons
, 2009
"... This paper describes an algorithm for generating a guaranteed intersectionfree interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of t ..."
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Cited by 7 (2 self)
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This paper describes an algorithm for generating a guaranteed intersectionfree interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against selfintersection is independent from a distance metric that determines the overall character of the interpolation sequence. This decoupled approach provides a powerful control mechanism for determining how the interpolation should appear, while still assuring against intersection and guaranteeing termination of the algorithm. Our algorithm also allows additional control by accommodating a set of algebraic constraints that can be weakly enforced throughout the interpolation sequence.
More Classes of Stuck Unknotted Hexagons
, 2003
"... Consider a hexagonal unknot with edges of xed length, for which we allow universal joint motions but do not allow edge crossings. We consider the maximum number of embedding classes that any such unknot may have. Until now, ve was a lower bound for this number. ..."
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Consider a hexagonal unknot with edges of xed length, for which we allow universal joint motions but do not allow edge crossings. We consider the maximum number of embedding classes that any such unknot may have. Until now, ve was a lower bound for this number.