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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 14 (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.
A Sum of Squares Theorem for Visibility Complexes and Applications
, 2001
"... We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses sim ..."
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Cited by 12 (1 self)
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We present a new method to implement in constant amortized time the ip operation of the socalled Greedy Flip Algorithm, an optimal algorithm to compute the visibility graph or the visibility complex of a collection of pairwise disjoint bounded convex sets of constant complexity. The method uses simple data structures and only the leftturn or counterclockwise predicate; it relies, among other things, on a sum of squares like theorem for visibility complexes stated and proved in this paper. (The sum of squares theorem for a simple arrangement of lines states that the average value of the square of the number of vertices of a face of the arrangement is a O(1).)
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 12 (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.
Deformable freespace tilings for kinetic collision detection
 I. J. Robotic Res
, 2002
"... We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic s ..."
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Cited by 11 (0 self)
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We present kinetic data structures for detecting collisions between a set of polygons that are moving continuously. Unlike classical collision detection methods that rely on bounding volume hierarchies, our method is based on deformable tilings of the free space surrounding the polygons. The basic shape of our tiles is that of a pseudotriangle, a shape sufficiently flexible to allow extensive deformation, yet structured enough to make detection of selfcollisions easy. We show different schemes for maintaining pseudotriangulations as a kinetic data structure, and we analyze their performance. Specifically, we first describe an algorithm for maintaining a pseudotriangulation of a point set, and show that the pseudotriangulation changes only quadratically many times if points move along algebraic arcs of constant degree. In addition, by refining the pseudotriangulation, we show triangulations of points that only change about O(n 7/3) times for linear motion. We then describe an algorithm for maintaining a pseudotriangulation of a set of convex polygons. Finally, we extend our algorithm to the general case of maintaining a pseudotriangulation of a set of moving or deforming simple polygons.
Combinatorial Roadmaps in Configuration Spaces of Simple Planar Polygons
"... Onedegreeoffreedom mechanisms induced by minimum pseudotriangulations with one convex hull edge removed have been recently introduced by the author to solve a family of noncolliding motion planning problems for planar robot arms (open or closed polygonal chains). They induce canonical roadmaps ..."
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Cited by 10 (4 self)
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Onedegreeoffreedom mechanisms induced by minimum pseudotriangulations with one convex hull edge removed have been recently introduced by the author to solve a family of noncolliding motion planning problems for planar robot arms (open or closed polygonal chains). They induce canonical roadmaps in configuration spaces of simple planar polygons with fixed edge lengths. While the combinatorial part is well understood, the search for efficient solutions to the algebraic components of the algorithm is posing a number of interesting questions, some of which are addressed in this paper. A list of open problems and further research topics on pointed pseudotriangulations and related structures motivated by this work is appended.
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 9 (8 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.
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 9 (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
On constrained minimum pseudotriangulations
 COMPUTING AND COMBINATORICS, PROC. 9TH ANN. INTERN. COMPUTING AND COMBINATORICS CONF. (COCOON 2003), VOLUME 2697 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2003
"... In this paper, we show some properties of a pseudotriangle and present three combinatorial bounds: the ratio of the size of minimum pseudotriangulation of a point set S and the size of minimal pseudotriangulation contained in a triangulation T, the ratio of the size of the best minimal pseudotriang ..."
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Cited by 9 (2 self)
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In this paper, we show some properties of a pseudotriangle and present three combinatorial bounds: the ratio of the size of minimum pseudotriangulation of a point set S and the size of minimal pseudotriangulation contained in a triangulation T, the ratio of the size of the best minimal pseudotriangulation and the worst minimal pseudotriangulation both contained in a given triangulation T, and the maximum number of edges in any settings of S and T. We also present a lineartime algorithm for finding a minimal pseudotriangulation contained in a given triangulation. We finally study the minimum pseudotriangulation containing a given set of noncrossing line segments.
Segment Endpoint Visibility Graphs are Hamiltonian
 COMPUT. GEOM
, 2002
"... We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian. ..."
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Cited by 8 (3 self)
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We show that the segment endpoint visibility graph of any finite set of disjoint line segments in the plane admits a simple Hamiltonian polygon, if not all segments are collinear. This proves a conjecture of Mirzaian.
Enumerating PseudoTriangulations in the Plane
 In Proc. 14th Canad. Conf. Comp. Geom
, 2002
"... A pseudotriangle is a simple polygon with exactly three convex vertices. A minimum pseudotriangulation of a set S is a partition of the convex hull of S into the least number of interior disjoint pseudotriangles whose vertices are the points of S. The graph of pseudotriangulations G_pse has pseu ..."
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Cited by 7 (0 self)
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A pseudotriangle is a simple polygon with exactly three convex vertices. A minimum pseudotriangulation of a set S is a partition of the convex hull of S into the least number of interior disjoint pseudotriangles whose vertices are the points of S. The graph of pseudotriangulations G_pse has pseudotriangulations as vertices and flips as edges. We show that G_pse is connected and present an algorithm for enumerating minimum pseudotriangulations in O(log n) time per pseudotriangulation.