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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.
The polytope of noncrossing graphs on a planar point set
, 2003
"... For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “noncrossing marked graphs” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of noncr ..."
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Cited by 11 (5 self)
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For any finite set A of n points in R 2, we define a (3n − 3)dimensional simple polyhedron whose face poset is isomorphic to the poset of “noncrossing marked graphs” with vertex set A, where a marked graph is defined as a geometric graph together with a subset of its vertices. The poset of noncrossing graphs on A appears as the complement of the star of a face in that polyhedron. The polyhedron has a unique maximal bounded face, of dimension 2ni + n − 3 where ni is the number of points of A in the interior of conv(A). The vertices of this polytope are all the pseudotriangulations of A, and the edges are flips of two types: the traditional diagonal flips (in pseudotriangulations) and the removal or insertion of a single edge. As a byproduct of our construction we prove that all pseudotriangulations are infinitesimally rigid graphs.
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
Proportional Contact Representations of Planar Graphs
"... Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex w ..."
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Cited by 7 (5 self)
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Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2segment graphs, which include maximal outerplanar graphs and partial 2trees. 1
Rectangle and Square Representations of Planar Graphs
"... In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the recta ..."
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Cited by 5 (3 self)
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In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part we ask for representations by a dissections of a rectangle into squares. We
Decompositions, partitions, and coverings with convex polygons and pseudotriangles
 GRAPHS AND COMBINATORICS
, 2007
"... We propose a novel subdivision of the plane that consists of both convex polygons and pseudotriangles. This pseudoconvex decomposition is significantly sparser than either convex decompositions or pseudotriangulations for planar point sets and simple polygons. We also introduce pseudoconvex part ..."
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Cited by 4 (2 self)
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We propose a novel subdivision of the plane that consists of both convex polygons and pseudotriangles. This pseudoconvex decomposition is significantly sparser than either convex decompositions or pseudotriangulations for planar point sets and simple polygons. We also introduce pseudoconvex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramseytype results concerning disjoint empty convex kgons in point sets.
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.
Mixed Volume Techniques for Embeddings of Laman Graphs
"... We use Bernstein’s Theorem [1] to obtain combinatorial bounds for the number of embeddings of Laman graph frameworks modulo rigid motions. For this, we study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. The bounds can easily be computed and ..."
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Cited by 3 (1 self)
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We use Bernstein’s Theorem [1] to obtain combinatorial bounds for the number of embeddings of Laman graph frameworks modulo rigid motions. For this, we study the mixed volume of suitable systems of polynomial equations obtained from the edge length constraints. The bounds can easily be computed and for some classes of graphs, the bounds are tight.