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64
Straightening polygonal arcs and convexifying polygonal cycles
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2000
"... Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles ..."
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Cited by 79 (30 self)
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Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewisedifferentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the wellstudied carpenter’s rule conjecture.
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?
Expansive motions and the polytope of pointed pseudotriangulations
 Discrete and Computational Geometry  The GoodmanPollack Festschrift, Algorithms and Combinatorics
, 2003
"... We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriang ..."
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Cited by 45 (15 self)
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We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriangulations of the point set and whose edges are flips of interior pseudotriangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an ngon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a byproduct a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of
Tight degree bounds for pseudotriangulations of points
, 2003
"... We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this p ..."
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Cited by 33 (11 self)
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We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudotriangulation has at most four vertices). Both degree bounds are tight. Minimum pseudotriangulations realizing these bounds (individually but not jointly) can be constructed in O(n log n) time.
Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
Counting Triangulations and PseudoTriangulations of Wheels
 IN PROC. 13TH CANAD. CONF. COMPUT. GEOM
, 2001
"... Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of trian ..."
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Cited by 21 (5 self)
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Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of triangulations and pseudotriangulations vary depending on the placement of the interior point, their difference is always the (n2)nd Catalan number. We also prove an inequality #PT # 3 i #T for the numbers of minimum pseudotriangulations and triangulations of any point configuration with i interior points.
Geometric and Computational Aspects of Polymer Reconfiguration
, 2000
"... We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edg ..."
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Cited by 20 (3 self)
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We examine a few computational geometric problems concerning the structures of polymers. We use a standard model of a polymer, a polygonal chain (path of line segments) in three dimensions. The chain can be reconfigured in any manner as long as the edge lengths and the angles between consecutive edges remain fixed, and no two edges cross during the motion. We discuss preliminary results on the following problems. Given a chaib, select...
Recent Results in Computational Origami
 In Proceedings of the 3rd International Meeting of Origami Science, Math, and Education
, 2001
"... Computational origami is a recent branch of computer science studying efficient algorithms for solving paperfolding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown signi ..."
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Cited by 19 (3 self)
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Computational origami is a recent branch of computer science studying efficient algorithms for solving paperfolding problems. This field essentially began with Robert Lang's work on algorithmic origami design [25], starting around 1993. Since then, the field of computational origami has grown significantly. The purpose of this paper is to survey the work in the field, with a focus on recent results, and to present several open problems that remain. The survey cannot hope to be complete, but we attempt to cover most areas of interest.
The zigzag path of a pseudotriangulation
 In Proc. 8th International Workshop on Algorithms and Data Structures (WADS
, 2003
"... We define the path of a pseudotriangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divideandconquer type of approaches for suitable (i.e. decomposable) problems on pseudotriangulations. We illustrate this method by presenting a n ..."
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Cited by 16 (5 self)
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We define the path of a pseudotriangulation, a data structure generalizing the path of a triangulation of a point set. This structure allows us to use divideandconquer type of approaches for suitable (i.e. decomposable) problems on pseudotriangulations. We illustrate this method by presenting a novel algorithm that counts the number of pseudotriangulations of a point set. 1
Infinitesimally locked selftouching linkages with applications to locked trees
 Physical Knots: Knotting, Linking, and Folding of Geometric Objects in 3space
, 2002
"... Abstract. Recently there has been much interest in linkages (barandjoint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether ..."
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Cited by 16 (10 self)
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Abstract. Recently there has been much interest in linkages (barandjoint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether planar linkages are locked in many cases of interest. The idea is to examine selftouching or degenerate frameworks in which multiple edges converge to geometrically overlapping configurations. We show how to study whether such frameworks are locked using techniques from rigidity theory, in particular firstorder rigidity and equilibrium stresses. Then we show how to relate locked selftouching frameworks to locked frameworks that closely approximate the selftouching frameworks. Our motivation is that most existing approaches to locked linkages are based on approximations to selftouching frameworks. In particular, we show that a previously proposed locked tree in the plane [BDD + 02] can be easily proved locked using our techniques, instead of the tedious arguments required by standard analysis. We also present a new