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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 29 (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.
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
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 13 (4 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.
On the number of plane graphs
 PROC. 17TH ANN. ACMSIAM SYMP. ON DISCRETE ALGORITHMS
, 2006
"... We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extre ..."
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Cited by 8 (1 self)
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We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extremal configuration, the socalled double zigzag chain. Most noteworthy this example bears Θ ∗ ( √ 72 n) = Θ ∗ (8.4853 n) triangulations and Θ ∗ (41.1889 n) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.
On the number of pseudotriangulations of certain point sets
 J. Combin. Theory Ser. A
, 2007
"... We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significant ..."
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Cited by 8 (2 self)
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We pose a monotonicity conjecture on the number of pseudotriangulations of any planar point set, and check it on two prominent families of point sets, namely the socalled double circle and double chain. The latter has asymptotically 12 n n Θ(1) pointed pseudotriangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far. ⋆ Parts of this work were done while the authors visited the Departament de
NonCrossing Frameworks with NonCrossing Reciprocals
, 2004
"... We study noncrossing frameworks in the plane for which the classical reciprocal on the dual graph is also noncrossing. We give a complete description of the selfstresses on noncrossing frameworks G whose reciprocals are noncrossing, in terms of: the types of faces (only pseudotriangles and ps ..."
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Cited by 7 (3 self)
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We study noncrossing frameworks in the plane for which the classical reciprocal on the dual graph is also noncrossing. We give a complete description of the selfstresses on noncrossing frameworks G whose reciprocals are noncrossing, in terms of: the types of faces (only pseudotriangles and pseudoquadrangles are allowed); the sign patterns in the stress on G; and a geometric condition on the stress vectors at some of the vertices. As in other recent papers where the interplay of noncrossingness and rigidity of straightline plane graphs is studied, pseudotriangulations show up as objects of special interest. For example, it is known that all planar Laman circuits can be embedded as a pseudotriangulation with one nonpointed vertex. We show that for such pseudotriangulation embeddings of planar Laman circuits which are sufficiently generic, the reciprocal is noncrossing and again a pseudotriangulation embedding of a planar Laman circuit. For a singular (nongeneric) pseudotriangulation embedding of a planar Laman circuit, the reciprocal is still noncrossing and a pseudotriangulation, but its underlying graph may not
Pseudotriangulations, rigidity and motion planning
 Discrete and Computational Geometry
, 2005
"... Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away fr ..."
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Cited by 3 (0 self)
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Abstract We propose a combinatorial approach to planning noncolliding trajectories for a polygonal barandjoint framework with n vertices. It is based on a new class of simple motionsinduced by expansive onedegreeoffreedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointedpseudotriangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties.The main application is an efficient algorithm for the Carpenter's Rule Problem: convexify a simple barandjoint planar polygonal linkage using only nonselfintersecting planarmotions. A step of the algorithm consists in moving a pseudotriangulationbased mechanism along its unique trajectory in configuration space until two adjacent edges align. At thealignment event, a local alteration restores the pseudotriangulation. The motion continues for O(n3) steps until all the points are in convex position. 1 Introduction We present a combinatorial solution to the Carpenter's Rule Problem: how to plan noncolliding reconfigurations of a planar robot arm. The main result is an efficient algorithm for the problem of continuously moving a simple planar polygon to any other configuration with the same edgelengths and orientation, while remaining in the plane and never creating selfintersections along the way. This is done by first finding motions that convexify both configurations with expansive motions (which never bring two points closer together) and then taking one path in reverse. All of the constructions are elementary and are based on a novel class of planar embedded graphs called pointed pseudotriangulations, for which we prove a variety of combinatorial and rigidity theoretical properties. More prominently, a pointed pseudotriangulation with a removed convex hull edge is a onedegreeoffreedom expansive mechanism. If its edges are seen as rigid bars (maintaining their lengths) and are allowed to rotate freely around the vertices (joints), the mechanism follows (for a well defined, finite time interval) a continuous trajectory along which no distance between a pair of points ever decreases. The expansive motion induced by these mechanisms provide the building blocks of our algorithm.
Planar pseudotriangulations, spherical pseudotilings and hyperbolic virtual polytopes
, 2006
"... Abstract. We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudotriangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other h ..."
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Cited by 3 (0 self)
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Abstract. We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of pseudotriangulations which was useful for implicit solution of the carpenter’s rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov’s problem): suppose that a constant C separates (nonstrictly) everywhere the principal curvature radii of a smooth 3dimensional convex body K. Then K is necessarily a ball of radius C. The two key ideas are: • Passing from planar pseudotriangulations to spherical pseudotilings, we avoid nonpoited vertices. Instead, we use pseudodigons. A theorem on spherically embedded Lamanplusone graphs is announced. • The difficult problem of hyperbolic polytopes constructing can be reduced
COMBINATORIAL PSEUDOTRIANGULATIONS
, 2005
"... Abstract. We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudotriangulation. This generalizes the main result of [4] which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudotriangulation, CPT, in t ..."
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Abstract. We prove that a planar graph is generically rigid in the plane if and only if it can be embedded as a pseudotriangulation. This generalizes the main result of [4] which treats the minimally generically rigid case. The proof uses the concept of combinatorial pseudotriangulation, CPT, in the plane and has two main steps: showing that a certain “generalized Laman property ” is a necessary and sufficient condition for a CPT to be “stretchable”, and showing that all generically rigid plane graphs admit a CPT assignment with that property. Additionally, we propose the study of combinatorial pseudotriangulations on closed surfaces. 1.