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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
A Combinatorial Approach to Planar Noncolliding Robot Arm Motion Planning
 In Proc. 41st FOCS
, 2000
"... We propose a combinatorial approach to plan noncolliding motions for a planar robot arm. The approach works even with certain types of movable polygonal obstacles and flexible polygonal fences. This yields a very efficient deterministic algorithm for a category of robot arm motion planning problems ..."
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Cited by 100 (14 self)
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We propose a combinatorial approach to plan noncolliding motions for a planar robot arm. The approach works even with certain types of movable polygonal obstacles and flexible polygonal fences. This yields a very efficient deterministic algorithm for a category of robot arm motion planning problems with many degrees of freedom, for which the known general roadmap techniques have exponential complexity. The main result is an efficient algorithm for convexifying a simple (open or closed) polygonal path with rigid nonintersecting motions in the plane. It works by computing in O(n²) time a monotone mechanism with one degree of freedom, whose motion is controlled by the rotation of a single edge around one of its endpoints. As it moves, all the interdistances between pairs of points not joined by a bar are nondecreasing, thus guaranteeing noncollision. At most O(n²) such motions suffice to reach a convex configuration of the original linkage. At each step, recomputing the next motion from ...
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
Simultaneous Edge Flipping in Triangulations
, 2003
"... We generalize the operation of flipping an edge in a triangulation to that of flipping several edges simultaneously. Our main result is an optimal upper bound on the number of parallel flips that are needed to transform a triangulation into another. Our results hold for triangulations of point se ..."
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Cited by 20 (5 self)
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We generalize the operation of flipping an edge in a triangulation to that of flipping several edges simultaneously. Our main result is an optimal upper bound on the number of parallel flips that are needed to transform a triangulation into another. Our results hold for triangulations of point sets and for polygons. 1 Introduction Given a triangulation T of a set P of points in the plane, an edge e of T is flippable if it is incident to two triangles whose union is a convex quadrilateral C. By flipping e we mean the operation of removing e from T and replacing it by the other diagonal of C. In this way we obtain a new triangulation T 0 of P , and we say that T 0 has been obtained from T by means of a flip. Local transformations on triangulations such as flips have been used in various fields starting with a simple greedy algorithm by Lawson that constructs the Delaunay triangulation of a point set by succesive flips from an arbitrary initial triangulation of the point set (se...
TwoPoint Euclidean Shortest Path Queries in the Plane (Extended Abstract)
, 1999
"... ) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the pla ..."
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Cited by 18 (2 self)
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) To appear in Proc. Tenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA '99), January 1719, 1999 YiJen Chiang Joseph S. B. Mitchell y Abstract We consider the twopoint query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s; t), of an Euclidean shortest obstacleavoiding path, ß(s; t), from s to t. Additionally, our data structure should allow one to report the path ß(s; t), in time proportional to its (combinatorial) size. We present various methods for solving this twopoint query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log 2 n) or optimal O(log n) query times, using polynomialspace data structures, with various tradeoffs between space and query time. While several results have been known for approximate twopoint Euclidean shortest p...
A Computational Geometric Approach To Visual Hulls
, 1997
"... Recognizing 3D objects from their 2D silhouettes is a popular topic in computer vision. Object reconstruction can be performed using the volume intersection approach. The visual hull of an object is the best approximation of an object that can be obtained by volume intersection. From the point of vi ..."
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Cited by 16 (0 self)
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Recognizing 3D objects from their 2D silhouettes is a popular topic in computer vision. Object reconstruction can be performed using the volume intersection approach. The visual hull of an object is the best approximation of an object that can be obtained by volume intersection. From the point of view of recognition from silhouettes, the visual hull can not be distinguished from the original object. In this paper, we present efficient algorithms for computing visual hulls. We start with the case of planar figures (polygons and curved objects) and base our approach on an efficient algorithm for computing the visibility graph of planar figures. We present and tackle many topics related to the query of visual hulls and to the recognition of objects equal to their visual hulls. We then move on to the 3dimensional case and give a flavor of how it may be approached. Keywords: Object reconstruction, volume intersection, visual hulls, visibility graphs, visibility complexes 1. Introduction ...
Radiosity for Dynamic Scenes in Flatland With the Visibility Complex
, 1996
"... The radiosity method is particularly suitable for global illumination calculations in static environments. Nonetheless, recent applications of image synthesis such as architectural simulation or lighting design require the ability to modify environments. Previous methods have attempted to deal with ..."
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Cited by 15 (2 self)
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The radiosity method is particularly suitable for global illumination calculations in static environments. Nonetheless, recent applications of image synthesis such as architectural simulation or lighting design require the ability to modify environments. Previous methods have attempted to deal with dynamic environments (environments where the geometry, the material properties, etc., can change) but still suffer some limitations in the case of moving objects. One of the main problems remaining is the efficient and accurate detection of which form factors must really be recomputed, since their calculation is the most timeconsuming part of the radiosity method. To correctly understand and solve this problem, we start with a method in 2D for polygonal scenes using the visibility complex. It is a powerful data structure representing the visibility relationships between objects in the plane. We have developed and implemented an algorithm which uses this structure to efficiently compute the ...
On Local Transformation of Polygons with Visibility Properties
 Theoretical Computer Science
, 2002
"... One strategy for the enumeration of a class of objects is local transformation, in which new objects of the class are produced by means of a small modification of a previouslyvisited object in the same class. When local transformation is possible, the operation can be used to generate objects of th ..."
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Cited by 13 (2 self)
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One strategy for the enumeration of a class of objects is local transformation, in which new objects of the class are produced by means of a small modification of a previouslyvisited object in the same class. When local transformation is possible, the operation can be used to generate objects of the class via random walks, and as the basis for such optimization heuristics as simulated annealing. For general simple polygons on fixed point sets, it is still not known whether the class of polygons on the set is connected via a constantsize local transformation. In this paper, we exhibit a simple local transformation for which the following polygon classes are connected: monotone, xmonotone, starshaped, (weakly) edgevisible and (weakly) externally visible. The latter class is particularly interesting as it is the most general polygon class known to be connected under local transformation.
Using the Visibility Complex for Radiosity Computation
 In ACM Workshop Appl. Comput. Geom
, 1996
"... The radiosity method is particularly suitable for global illumination calculations in static environments. Nonetheless, for applications of image synthesis such as lighting design or architectural simulation, we have to deal with dynamic environments. To make the method usable in a real case, the il ..."
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Cited by 10 (2 self)
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The radiosity method is particularly suitable for global illumination calculations in static environments. Nonetheless, for applications of image synthesis such as lighting design or architectural simulation, we have to deal with dynamic environments. To make the method usable in a real case, the illumination has to be updated as fast as possible after an object moves. The e#cient way is to find the calculations strictly necessary to be recomputed after a change in the scene. The largest part of the computation time is spent on visibility calculation. In this paper, we investigate the possible speed ups in those calculations. We propose the use of the visibility complex for radiosity calculations. The presented study is realized for 2D scenes of convex objects in the static case. We show that the visibility complex is very suitable for radiosity calculations in this context, and that it also allows for e#cient updates in the dynamic case. Keywords: radiosity, discontinuity meshing, for...