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Geometric Shortest Paths and Network Optimization
 Handbook of Computational Geometry
, 1998
"... Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of t ..."
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Cited by 147 (12 self)
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Introduction A natural and wellstudied problem in algorithmic graph theory and network optimization is that of computing a "shortest path" between two nodes, s and t, in a graph whose edges have "weights" associated with them, and we consider the "length" of a path to be the sum of the weights of the edges that comprise it. Efficient algorithms are well known for this problem, as briefly summarized below. The shortest path problem takes on a new dimension when considered in a geometric domain. In contrast to graphs, where the encoding of edges is explicit, a geometric instance of a shortest path problem is usually specified by giving geometric objects that implicitly encode the graph and its edge weights. Our goal in devising efficient geometric algorithms is generally to avoid explicit construction of the entire underlying graph, since the full induced graph may be very large (even exponential in the input size, or infinite). Computing an optimal
Deformable free space tilings for kinetic collision detection
 International Journal of Robotics Research
, 2000
"... We present kinetic data structures for detecting collisions between a set of polygons that are not only moving continuously but whose shapes can also change continuously with time. We construct a planar subdivision of the common exterior of the polygons, called a pseudotriangulation, that certifies ..."
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Cited by 78 (13 self)
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We present kinetic data structures for detecting collisions between a set of polygons that are not only moving continuously but whose shapes can also change continuously with time. We construct a planar subdivision of the common exterior of the polygons, called a pseudotriangulation, that certifies their disjointness. 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. We then describe an algorithm for maintaining a pseudotriangulation of a set of convex polygons. Finally, we extend our algorithm to maintaining a pseudotriangulation of a set of simple polygons.
Expansive motions and the polytope of pointed pseudotriangulations
 Discrete and Computational Geometry  The GoodmanPollack Festschrift, Algorithms and Combinatorics
, 2003
"... We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriang ..."
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Cited by 45 (15 self)
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We introduce the polytope of pointed pseudotriangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1skeleton is the graph whose vertices are the pointed pseudotriangulations of the point set and whose edges are flips of interior pseudotriangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an ngon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a byproduct a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of
Tight degree bounds for pseudotriangulations of points
, 2003
"... We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this p ..."
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Cited by 33 (11 self)
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We show that every set of n points in general position has a minimum pseudotriangulation whose maximum vertex degree is five. In addition, we demonstrate that every point set in general position has a minimum pseudotriangulation whose maximum face degree is four (i.e., each interior face of this pseudotriangulation has at most four vertices). Both degree bounds are tight. Minimum pseudotriangulations realizing these bounds (individually but not jointly) can be constructed in O(n log n) time.
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 (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.
Counting Triangulations and PseudoTriangulations of Wheels
 IN PROC. 13TH CANAD. CONF. COMPUT. GEOM
, 2001
"... Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of trian ..."
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Cited by 21 (5 self)
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Motivated by several open questions on triangulations and pseudotriangulations, we give closed form expressions for the number of triangulations and the number of minimum pseudotriangulations of n points in wheel configurations, that is, with n  1 in convex position. Although the numbers of triangulations and pseudotriangulations vary depending on the placement of the interior point, their difference is always the (n2)nd Catalan number. We also prove an inequality #PT # 3 i #T for the numbers of minimum pseudotriangulations and triangulations of any point configuration with i interior points.
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
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.
Interactive Ray Tracing With the Visibility Complex
, 1999
"... We describe a method of producing raytraced images of 2D environments at interactive rates. The 2D environment consists of a set of disjoint, convex polygons. Our technique is based on the visibility complex [17,19] [Pocchiola M, Vegter G. Proc Int J Comput GEOM Applic 1996;6(3):279}308. Rivie re S ..."
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Cited by 14 (0 self)
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We describe a method of producing raytraced images of 2D environments at interactive rates. The 2D environment consists of a set of disjoint, convex polygons. Our technique is based on the visibility complex [17,19] [Pocchiola M, Vegter G. Proc Int J Comput GEOM Applic 1996;6(3):279}308. Rivie re S. Visibility computations in 2D polygonal scenes. PhD thesis, Univ. Joseph Fourier, Grenoble I, France], a data structure in a dual space where a face of the visibility complex corresponds to a contiguous set of rays in the primary space with the same forward and backward views. Sweeping the viewing ray around a viewpoint corresponds to walking along a trajectory on the visibility complex. Producing a raytraced image is equivalent to walking along and maintaining a set of trajectories. Generating raytraced images with the visibility complex is very e$cient since it uses the coherence among the rays e!ectively. We have developed a new algorithm for the randomized incremental construction of the visibility complex. The advantage of using an incremental algorithm is that the history of the incremental construction yields an e$cient rayquery data structure, which is required for casting secondary rays. The performance of our algorithm is analyzed and a comparison is made with the classical raytracing algorithm. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Interactive ray tracing; Visibility complex; Coherence; Line space 1.