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127
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 560 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
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.
The Power Crust
, 2001
"... The power crust is a construction which takes a sample of points from the surface of a threedimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce ..."
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Cited by 201 (6 self)
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The power crust is a construction which takes a sample of points from the surface of a threedimensional object and produces a surface mesh and an approximate medial axis. The approach is to first approximate the medial axis transform (MAT) of the object. We then use an inverse transform to produce the surface representation from the MAT.
Voronoi Diagrams and Delaunay Triangulations
 Computing in Euclidean Geometry
, 1992
"... The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi ..."
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Cited by 198 (3 self)
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The Voronoi diagram is a fundamental structure in computationalgeometry and arises naturally in many different fields. This chapter surveys properties of the Voronoi diagram and its geometric dual, the Delaunay triangulation. The emphasis is on practical algorithms for the construction of Voronoi diagrams. 1 Introduction Let S be a set of n points in ddimensional euclidean space E d . The points of S are called sites. The Voronoi diagram of S splits E d into regions with one region for each site, so that the points in the region for site s2S are closer to s than to any other site in S. The Delaunay triangulation of S is the unique triangulation of S so that there are no elements of S inside the circumsphere of any triangle. Here `triangulation' is extended from the planar usage to arbitrary dimension: a triangulation decomposes the convex hull of S into simplices using elements of S as vertices. The existence and uniqueness of the Delaunay triangulation are perhaps not obvio...
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 180 (8 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Spanning Trees and Spanners
, 1996
"... We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. ..."
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Cited by 143 (2 self)
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We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and lowdilation graphs. 1 Introduction This survey covers topics in geometric network design theory. The problem is easy to state: connect a collection of sites by a "good" network. For instance, one may wish to connect components of a VLSI circuit by networks of wires, in a way that uses little surface area on the chip, draws little power, and propagates signals quickly. Similar problems come up in other applications such as telecommunications, road network design, and medical imaging [1]. One network design problem, the Traveling Salesman problem, is sufficiently important to have whole books devoted to it [79]. Problems involving some form of geometric minimum or maximum spanning tree also arise in the solution of other geometric problems such as clustering [12], mesh generation [56], and robot motion planning [93]. One can vary the network design problem in many w...
Fast Polygonal Approximation of Terrains and Height Fields
, 1995
"... Several algorithms for approximating terrains and other height fields using polygonal meshes are described, compared, and optimized. These algorithms take a height field as input, typically a rectangular grid of elevation data H(x; y), and approximate it with a mesh of triangles, also known as a tri ..."
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Cited by 142 (5 self)
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Several algorithms for approximating terrains and other height fields using polygonal meshes are described, compared, and optimized. These algorithms take a height field as input, typically a rectangular grid of elevation data H(x; y), and approximate it with a mesh of triangles, also known as a triangulated irregular network, or TIN. The algorithms attempt to minimize both the error and the number of triangles in the approximation. Applications include fast rendering of terrain data for flight simulation and fitting of surfaces to range data in computer vision. The methods can also be used to simplify multichannel height fields such as textured terrains or planar color images. The most successful method we examine is the greedy insertion algorithm. It begins with a simple triangulation of the domain and, on each pass, finds the input point with highest error in the current approximation and inserts it as a vertex in the triangulation. The mesh is updated either with Delaunay triangul...
Four Results on Randomized Incremental Constructions
 Comput. Geom. Theory Appl
, 1993
"... We prove four results on randomized incremental constructions (RICs): ffl an analysis of the expected behavior under insertion and deletions, ffl a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, ffl a tail estimate for the space complexity of RICs, ffl a lower ..."
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Cited by 92 (17 self)
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We prove four results on randomized incremental constructions (RICs): ffl an analysis of the expected behavior under insertion and deletions, ffl a fully dynamic data structure for convex hull maintenance in arbitrary dimensions, ffl a tail estimate for the space complexity of RICs, ffl a lower bound on the complexity of a game related to RICs. 1
Learning Maps for Indoor Mobile Robot Navigation
 ARTIFICIAL INTELLIGENCE (ACCEPTED FOR PUBLICATION)
, 1997
"... Autonomous robots must be able to learn and maintain models of their environments. Research on mobile robot navigation has produced two major paradigms for mapping indoor environments: gridbased and topological. While gridbased methods produce accurate metric maps, their complexity often prohibits ..."
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Cited by 83 (12 self)
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Autonomous robots must be able to learn and maintain models of their environments. Research on mobile robot navigation has produced two major paradigms for mapping indoor environments: gridbased and topological. While gridbased methods produce accurate metric maps, their complexity often prohibits efficient planning and problem solving in largescale indoor environments. Topological maps, on the other hand, can be used much more efficiently, yet accurate and consistent topological maps are often difficult to learn and maintain in largescale environments, particularly if momentary sensor data is highly ambiguous. This paper describes an approach that integrates both paradigms: gridbased and topological. Gridbased maps are learned using artificial neural networks and naive Bayesian integration. Topological maps are generated on top of the gridbased maps, by partitioning the latter into coherent regions. By combining both paradigms, the approach presented here gains advantages from both worlds: accuracy/consistency and efficiency. The paper gives results for autonomous exploration, mapping and operation of a mobile robot in populated multiroom environments.