Results 1  10
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17
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 (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.
Convexity Minimizes PseudoTriangulations
 Computational Geometry: Theory and Applications
, 2002
"... The number of minimum pseudotriangulations is minimized for point sets in convex position. ..."
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Cited by 15 (2 self)
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The number of minimum pseudotriangulations is minimized for point sets in convex position.
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.
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 11 (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).)
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.
Enumerating pseudotriangulations in the plane
 COMPUT. GEOM. THEORY APPL
, 2005
"... A pseudotriangle is a simple polygon with exactly three convex vertices. A pseudotriangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudotriangles whose vertices are points of S. A pointed pseudotriangulation is one which has the le ..."
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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 pseudotriangulation of a finite point set S in the plane is a partition of the convex hull of S into interior disjoint pseudotriangles whose vertices are points of S. A pointed pseudotriangulation is one which has the least number of pseudotriangles. We study the graph G whose vertices represent the pointed pseudotriangulations and whose edges represent flips. We present an algorithm for
Degree Bounds for Constrained Pseudotriangulations
"... We introduce the concept of a constrained pointed pseudotriangulation T G of a point set S with respect to a pointed planar straight line graph G = (S, E). Forthe case that G forms a simple polygon P with vertex set S we give tight bounds on the vertex degree of TG. ..."
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Cited by 6 (0 self)
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We introduce the concept of a constrained pointed pseudotriangulation T G of a point set S with respect to a pointed planar straight line graph G = (S, E). Forthe case that G forms a simple polygon P with vertex set S we give tight bounds on the vertex degree of TG.
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 6 (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.
Pointed binary encompassing trees: simple and optimal
 COMPUT. GEOM. THEORY APPL
, 2007
"... For n disjoint line segments in the plane we construct in optimal O(n log n) time an encompassing tree of maximal degree three such that every vertex is pointed. Moreover, at every segment endpoint all incident edges lie in a halfplane defined by the incident input segment. ..."
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Cited by 4 (0 self)
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For n disjoint line segments in the plane we construct in optimal O(n log n) time an encompassing tree of maximal degree three such that every vertex is pointed. Moreover, at every segment endpoint all incident edges lie in a halfplane defined by the incident input segment.