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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 146 (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
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
An Efficient Algorithm for Euclidean Shortest Paths Among Polygonal Obstacles in the Plane
, 1988
"... We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time. ..."
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Cited by 34 (1 self)
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We give an algorithm to compute a (Euclidean) shortest path in a polygon with h holes and a total of n vertices. The algorithm uses O(n) space and requires O(n + h² log n) time.
Computing the Visibility Graph via Pseudotriangulations
 In Proc. 11th Annu. ACM Sympos. Comput. Geom
, 1995
"... We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k+n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudotriangulations, whose combinat ..."
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Cited by 31 (2 self)
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We show that the k free bitangents of a collection of n pairwise disjoint convex plane sets can be computed in time O(k+n log n) and O(n) working space. The algorithm uses only one advanced data structure, namely a splittable queue. We introduce (weakly) greedy pseudotriangulations, whose combinatorial properties are crucial for our method. 1 Introduction Consider a collection O of pairwise disjoint convex objects in the plane. We are interested in problems in which these objects arise as obstacles, either in connection with visibility problems where they can block the view from an other geometric object, or in motion planning, where these objects may prevent a moving object from moving along a straight line path. The visibility graph is a central object in such contexts. For polygonal obstacles the vertices of these polygons are the nodes of the visibility graph, and two nodes are connected by an arc if the corresponding vertices can see each other. [9] describes the first nontriv...
Exact FromRegion Visibility Culling
, 2002
"... To preprocess a scene for the purpose of visibility culling during walkthroughs it is necessary to solve visibility from all the elements of a finite partition of viewpoint space. Many conservative and approximate solutions have been developed that solve for visibility rapidly. The idealised exac ..."
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Cited by 28 (1 self)
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To preprocess a scene for the purpose of visibility culling during walkthroughs it is necessary to solve visibility from all the elements of a finite partition of viewpoint space. Many conservative and approximate solutions have been developed that solve for visibility rapidly. The idealised exact solution for general 3D scenes has often been regarded as computationally intractable. Our exact algorithm for finding the visible polygons in a scene from a region is a computationally tractable preprocess that can handle scenes of the order of millions of polygons. The essence
Pseudotriangulations: Theory and Applications
 In Proc. 12th Annu. ACM Sympos. Comput. Geom
, 1996
"... this paper is (1) to give three new applications of these concepts to 2dimensional visibility problems, and (2) to study realizability questions suggested by the pseudotrianglepseudoline duality; see Figure 1. Our first application is related to the rayshooting problem in the plane: preprocess a ..."
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Cited by 25 (4 self)
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this paper is (1) to give three new applications of these concepts to 2dimensional visibility problems, and (2) to study realizability questions suggested by the pseudotrianglepseudoline duality; see Figure 1. Our first application is related to the rayshooting problem in the plane: preprocess a set of objects into a data structure such that the first object hit by a query ray can be computed efficiently. In section 3 we show that for a scene of n objects, where the objects are pairwise disjoint convex sets with m 'simple' arcs in total, one can obtain O(log m) query time using
The VisibilityVoronoi complex and its applications
 In Proc. 21st Annu. ACM Sympos. Comput. Geom. (SCG
, 2005
"... We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 ..."
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Cited by 25 (3 self)
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We introduce a new type of diagram called the VV (c)diagram (the Visibility–Voronoi diagram for clearance c), which is a hybrid between the visibility graph and the Voronoi diagram of polygons in the plane. It evolves from the visibility graph to the Voronoi diagram as the parameter c grows from 0 to ∞. This diagram can be used for planning naturallooking paths for a robot translating amidst polygonal obstacles in the plane. A naturallooking path is short, smooth, and keeps — where possible — an amount of clearance c from the obstacles. The VV (c)diagram contains such paths. We also propose an algorithm that is capable of preprocessing a scene of configurationspace polygonal obstacles and constructs a data structure called the VVcomplex. The VVcomplex can be used to efficiently plan motion paths for any start and goal configuration and any clearance value c, without having to explicitly construct the VV (c)diagram for that cvalue. The preprocessing time is O(n 2 log n), where n is the total number of obstacle vertices, and the data structure can be queried directly for any cvalue by merely performing a Dijkstra search. We have implemented a Cgalbased software package for computing the VV (c)diagram in an exact manner for a given clearance value, and used it to plan naturallooking paths in various applications.
Distanceoptimal navigation in an unknown environment without sensing distances
 IEEE Transactions on Robotics
, 2007
"... Abstract — This paper considers what can be accomplished using a mobile robot that has limited sensing. For navigation and mapping, the robot has only one sensor, which tracks the directions of depth discontinuities. There are no coordinates, and the robot is given a motion primitive that allows it ..."
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Cited by 25 (13 self)
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Abstract — This paper considers what can be accomplished using a mobile robot that has limited sensing. For navigation and mapping, the robot has only one sensor, which tracks the directions of depth discontinuities. There are no coordinates, and the robot is given a motion primitive that allows it to move toward discontinuities. The robot is incapable of performing localization or measuring any distances or angles. Nevertheless, when dropped into an unknown planar environment, the robot builds a data structure, called the Gap Navigation Tree, which enables it to navigate optimally in terms of Euclidean distance traveled. In a sense, the robot is able to learn the critical information contained in the classical shortestpath roadmap, although surprisingly it is unable to extract metric information. We prove these results for the case of a point robot placed into a simply connected, piecewiseanalytic planar environment. The case of multiply connected environments is also addressed, in which it is shown that further sensing assumptions are needed. Due to the limited sensor given to the robot, globally optimal navigation is impossible; however, our approach achieves locally optimal (within a homotopy class) navigation, which is the best that is theoretically possible under this robot model. Index Terms — Visibility, navigation, optimality, map building, minimal sensing, shortest paths, information spaces, sensorbased
Lines and free line segments tangent to arbitrary threedimensional convex polyhedra
 SIAM Journal on Computing
, 2006
"... SUE WHITESIDES∗ ∗ Abstract. Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of threedimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in R 3 with a ..."
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Cited by 24 (14 self)
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SUE WHITESIDES∗ ∗ Abstract. Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of threedimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in R 3 with a total of n edges consists of Θ(n 2) connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of k possibly intersecting convex polyhedra with a total of n edges admits, in the worst case, Θ(n 2 k 2) connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an O(n 2 k 2 log n) time and O(nk 2) space algorithm that, given a scene of k possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines. Key words. computational geometry, 3D visibility, visibility complex, visual events