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54
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 87 (31 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.
RIGIDITY AND SCENE ANALYSIS
"... Rigidity and flexibility of frameworks (motions preserving lengths of bars) and scene analysis (liftings from plane polyhedral pictures to spatial polyhedra) are two core examples of a general class of geometric problems: (a) Given a discrete configuration of points,lines,planes,... in Euclidean spa ..."
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Cited by 48 (9 self)
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Rigidity and flexibility of frameworks (motions preserving lengths of bars) and scene analysis (liftings from plane polyhedral pictures to spatial polyhedra) are two core examples of a general class of geometric problems: (a) Given a discrete configuration of points,lines,planes,... in Euclidean space,
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 39 (15 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.
PseudoTriangulations  a Survey
 CONTEMPORARY MATHEMATICS
"... A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory an ..."
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Cited by 25 (5 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices, and a pseudotriangulation is a facetoface tiling of a planar region into pseudotriangles. Pseudotriangulations appear as data structures in computational geometry, as planar barandjoint frameworks in rigidity theory and as projections of locally convex surfaces. This survey of current literature includes combinatorial properties and counting of special classes, rigidity theoretical results, representations as polytopes, straightline drawings from abstract versions called combinatorial pseudotriangulations, algorithms and applications of pseudotriangulations.
Multitriangulations as complexes of star polygons
, 2007
"... Maximal (k+1)crossingfree graphs on a planar point set in convex position, that is, ktriangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at ktriangulations, namely as complexes of star poly ..."
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Cited by 24 (8 self)
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Maximal (k+1)crossingfree graphs on a planar point set in convex position, that is, ktriangulations, have received attention in recent literature, with motivation coming from several interpretations of them. We introduce a new way of looking at ktriangulations, namely as complexes of star polygons. With this tool we give new, direct, proofs of the fundamental properties of ktriangulations, as well as some new results. This interpretation also opensup new avenues of research, that we briefly explore in the last section.
Bijections for Baxter Families and Related Objects
, 2008
"... The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2) ..."
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Cited by 19 (8 self)
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The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)
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 ..."
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Cited by 18 (6 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.
Convexity Minimizes PseudoTriangulations
 COMPUTATIONAL GEOMETRY 28 (2004) 3–10
, 2004
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Multitriangulations, pseudotriangulations and primitive sorting networks
 Discrete Comput. Geom. (DOI
, 2012
"... Abstract. We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based ..."
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Cited by 15 (10 self)
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Abstract. We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.