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19
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?
On Triangulating ThreeDimensional Polygons
 COMPUTATIONAL GEOMETRY: THEORY AND APPLICATIONS
, 1996
"... A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficie ..."
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Cited by 29 (3 self)
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A threedimensional polygon is triangulable if it has a nonselfintersecting triangulation which defines a simplyconnected 2manifold. We show that the problem of deciding whether a 3dimensional polygon is triangulable is NPComplete. We then establish some necessary conditions and some sufficient conditions for a polygon to be triangulable, providing special cases when the decision problem may be answered in polynomial time.
Locked and Unlocked Polygonal Chains in 3D
, 1999
"... In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are main ..."
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Cited by 28 (14 self)
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In this paper, we study movements of simple polygonal chains in 3D. We say that an open, simple polygonal chain can be straightened if it can be continuously reconfigured to a straight sequence of segments in such a manner that both the length of each link and the simplicity of the chain are maintained throughout the movement. The analogous concept for closed chains is convexification: reconfiguration to a planar convex polygon. Chains that cannot be straightened or convexified are called locked. While there are open chains in 3D that are locked, we show that if an open chain has a simple orthogonal projection onto some plane, it can be straightened. For closed chains, we show that there are unknotted but locked closed chains, and we provide an algorithm for convexifying a planar simple polygon in 3D. All our algorithms require only O(n) basic "moves" and run in linear time.
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
Spaceefficient algorithms for approximating polygonal curves in twodimensional space
 Proc. 4th International Computing and Combinatorics Conf
, 1998
"... Given an nvertex polygonal curve P = [p 1, p 2, : ::, pn] in the 2dimensional space R 2, we consider the problem of approximating P by finding another polygonal curve P 0 ..."
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Cited by 23 (6 self)
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Given an nvertex polygonal curve P = [p 1, p 2, : ::, pn] in the 2dimensional space R 2, we consider the problem of approximating P by finding another polygonal curve P 0
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
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segme ..."
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Cited by 10 (3 self)
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We study straightline drawings of graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5n/2 segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
On Removing Nondegeneracy Assumptions in Computational Geometry (Extended Abstract)
, 1997
"... ) Francisco G'omez 1 , Suneeta Ramaswami 2 and Godfried Toussaint 2 1 Dept. of Applied Mathematics, Universidad Politecnica de Madrid, Madrid, Spain 2 School of Computer Science, McGill University, Montr'eal, Qu'ebec, Canada Abstract Existing methods for removing degeneracies in computation ..."
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Cited by 9 (7 self)
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) Francisco G'omez 1 , Suneeta Ramaswami 2 and Godfried Toussaint 2 1 Dept. of Applied Mathematics, Universidad Politecnica de Madrid, Madrid, Spain 2 School of Computer Science, McGill University, Montr'eal, Qu'ebec, Canada Abstract Existing methods for removing degeneracies in computational geometry can be classified as either approximation or perturbation methods. These methods give the implementer two rather unsatisfactory choices: find an approximate solution to the original problem given, or find an exact solution to an approximation of the original problem. We address an alternative approach that has received little attention in the computational geometry literature. Often a typical computational geometry paper will make a nondegeneracy assumption that can in fact be removed (without perturbing the input) by a global rigid transformation of the input. In these situations, by applying suitable pre and post processing steps to an algorithm, we obtain the exact soluti...
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