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22
The Quickhull algorithm for convex hulls
 ACM TRANSACTIONS ON MATHEMATICAL SOFTWARE
, 1996
"... The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algo ..."
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Cited by 456 (0 self)
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The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the twodimensional Quickhull Algorithm with the generaldimension BeneathBeyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floatingpoint arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.
Voronoi Diagrams
 Handbook of Computational Geometry
"... Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such t ..."
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Cited by 143 (19 self)
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Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.
A Comparison of Sequential Delaunay Triangulation Algorithms
, 1996
"... This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer’s divide and conquer algorithm, Fortune’s sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, an ..."
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Cited by 55 (0 self)
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This paper presents an experimental comparison of a number of different algorithms for computing the Deluanay triangulation. The algorithms examined are: Dwyer’s divide and conquer algorithm, Fortune’s sweepline algorithm, several versions of the incremental algorithm (including one by Ohya, Iri, and Murota, a new bucketingbased algorithm described in this paper, and Devillers’s version of a Delaunaytree based algorithm that appears in LEDA), an algorithm that incrementally adds a correct Delaunay triangle adjacent to a current triangle in a manner similar to gift wrapping algorithms for convex hulls, and Barber’s convex hull based algorithm. Most of the algorithms examined are designed for good performance on uniformly distributed sites. However, we also test implementations of these algorithms on a number of nonuniform distibutions. The experiments go beyond measuring total running time, which tends to be machinedependent. We also analyze the major highlevel primitives that algorithms use and do an experimental analysis of how often implementations of these algorithms perform each operation.
Improved Incremental Randomized Delaunay Triangulation
, 1997
"... We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location ..."
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Cited by 44 (9 self)
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We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, and small memory occupation. The location
The Delaunay hierarchy
 Internat. J. Found. Comput. Sci
"... We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into s ..."
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Cited by 36 (6 self)
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We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from that neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query.
On lazy randomized incremental construction
 In Proc. 26th Annu. ACM Sympos. Theory Comput
, 1994
"... We introduce a new type of randomized incremental algorithms. Contrary to standard randomized incremental algorithms, these lazy randomized incremental algorithms are suited for computing structures that have a `nonlocal' definition. In order to analyze these algorithms we generalize some results o ..."
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Cited by 32 (8 self)
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We introduce a new type of randomized incremental algorithms. Contrary to standard randomized incremental algorithms, these lazy randomized incremental algorithms are suited for computing structures that have a `nonlocal' definition. In order to analyze these algorithms we generalize some results on random sampling to such situations. We apply our techniques to obtain efficient algorithms for the computation of single cells in arrangements of segments in the plane, single cells in arrangements of triangles in space, and zones in arrangements of hyperplanes. We also prove combinatorial bounds on the complexity of what we call the (6k)cell in arrangements of segments in the plane or triangles in space; this is the set of all points on the segments (triangles) that can reach the origin with a path that crosses at most k, 1 segments (triangles).
Geometric Data Structures for Computer Graphics
, 2003
"... pefully make them curious about further powerful treasures to be discovered in the area of computational geometry. In order to achieve these goals in an engaging yet sound manner, the general concept throughout the course is to present each geometric data structure in the following way: first, th ..."
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Cited by 15 (3 self)
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pefully make them curious about further powerful treasures to be discovered in the area of computational geometry. In order to achieve these goals in an engaging yet sound manner, the general concept throughout the course is to present each geometric data structure in the following way: first, the data strucure will be defined and described in detail; then, some of its fundamental properties will be highlighted; after that, one or more computational geometry algorithms based on the data structure will be presented; and finally, a number of recent, representative and practically relevant algorithms from computer graphics will be described in detail, showing the utilization of the data structure in a creative and enlightening way. We have arranged the topics in roughly increasing degree of difficulty. The hierarchical data structures are ordered by increasing flexibility, while the nonhierarchical topics build on each other. Finally, the last topic presents a generic technique for
Randomization yields simple O(n log ∗ n) algorithms for difficult Ω(n) problems
 Internat. J. Comput. Geom. Appl
, 1992
"... We use here the results on the influence graph[1] to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(nlog n) to O(n log ⋆ n). This technique applies in the following applicatio ..."
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Cited by 15 (5 self)
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We use here the results on the influence graph[1] to adapt them for particular cases where additional information is available. In some cases, it is possible to improve the expected randomized complexity of algorithms from O(nlog n) to O(n log ⋆ n). This technique applies in the following applications: triangulation of a simple polygon, skeleton of a simple polygon, Delaunay triangulation of points knowing the EMST (euclidean minimum spanning tree).
Incremental Construction of the Delaunay Triangulation and the Delaunay Graph in Medium Dimension
, 2009
"... We describe a new implementation of the wellknown incremental algorithm for constructing Delaunay triangulations in any dimension. Our implementation follows the exact computing paradigm and is fully robust. Extensive comparisons show that our implementation outperforms the best currently available ..."
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Cited by 10 (1 self)
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We describe a new implementation of the wellknown incremental algorithm for constructing Delaunay triangulations in any dimension. Our implementation follows the exact computing paradigm and is fully robust. Extensive comparisons show that our implementation outperforms the best currently available codes for exact convex hulls and Delaunay triangulations, compares very well to the fast nonexact Qhull implementation and can be used for quite big input sets in spaces of dimensions up to 6. To circumvent prohibitive memory usage, we also propose a modi cation of the algorithm that uses and stores only the Delaunay graph (the edges of the full triangulation). We show that a careful implementation of the modi ed algorithm performs only 6 to 8 times slower than the original algorithm while drastically reducing memory usage in dimension 4 or above. 1
Dog Bites Postman: Point Location in the Moving Voronoi Diagram and Related Problems
, 1994
"... In this paper, we discuss two variations of the twodimensional postoffice problem that arise when the postoffices are n postmen moving with constant velocities. The first variation addresses the question: given a point q 0 and time t 0 who is the nearest postman to q 0 at time t 0 ?We present a ra ..."
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Cited by 6 (2 self)
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In this paper, we discuss two variations of the twodimensional postoffice problem that arise when the postoffices are n postmen moving with constant velocities. The first variation addresses the question: given a point q 0 and time t 0 who is the nearest postman to q 0 at time t 0 ?We present a randomized incremental data structure that answers the query in expected O(log² n) time. The second variation views a query point as a dog searching for a postman to bite and finds the postman that a dog running with speed v d could reach first. We show that if the dog is quicker than all of the postmen then the data structure developed for the first problem permits us to solve this one in O(log² n) time as well. The proposed