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19
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...
A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra
, 1992
"... We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following prope ..."
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Cited by 184 (28 self)
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We present a new piv otbased algorithm which can be used with minor modification for the enumeration of the facets of the convex hull of a set of points, or for the enumeration of the vertices of an arrangement or of a convex polyhedron, in arbitrary dimension. The algorithm has the following properties: (a) Virtually no additional storage is required beyond the input data; (b) The output list produced is free of duplicates; (c) The algorithm is extremely simple, requires no data structures, and handles all degenerate cases; (d) The running time is output sensitive for nondegenerate inputs; (e) The algorithm is easy to efficiently parallelize. For example, the algorithm finds the v vertices of a polyhedron in R d defined by a nondegenerate system of n inequalities (or dually, the v facets of the convex hull of n points in R d,where each facet contains exactly d given points) in time O(ndv) and O(nd) space. The v vertices in a simple arrangement of n hyperplanes in R d can be found in O(n 2 dv) time and O(nd) space complexity. The algorithm is based on inverting finite pivot algorithms for linear programming.
Iterated Nearest Neighbors and Finding Minimal Polytopes
, 1994
"... Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based o ..."
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Cited by 56 (6 self)
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Weintroduce a new method for finding several types of optimal kpoint sets, minimizing perimeter, diameter, circumradius, and related measures, by testing sets of the O(k) nearest neighbors to each point. We argue that this is better in a number of ways than previous algorithms, whichwere based on high order Voronoi diagrams. Our technique allows us for the first time to efficiently maintain minimal sets as new points are inserted, to generalize our algorithms to higher dimensions, to find minimal convex kvertex polygons and polytopes, and to improvemany previous results. Weachievemany of our results via a new algorithm for finding rectilinear nearest neighbors in the plane in time O(n log n+kn). We also demonstrate a related technique for finding minimum area kpoint sets in the plane, based on testing sets of nearest vertical neighbors to each line segment determined by a pair of points. A generalization of this technique also allows us to find minimum volume and boundary measure sets in arbitrary dimensions.
Lines in Space: Combinatorics and Algorithms
, 1996
"... Questions about lines in space arise frequently as subproblems in threedimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in threedimensional space. Our main results include: 1. A tight �(n2) b ..."
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Cited by 24 (4 self)
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Questions about lines in space arise frequently as subproblems in threedimensional computational geometry. In this paper we study a number of fundamental combinatorial and algorithmic problems involving arrangements of n lines in threedimensional space. Our main results include: 1. A tight �(n2) bound on the maximum combinatorial description complexity of the set of all oriented lines that have specified orientations relative to the n given lines. 2. A similar bound of �(n3) for the complexity of the set of all lines passing above the n given lines. 3. A preprocessing procedure using O(n2+ε) time and storage, for anyε>0, that builds a structure supporting O(log n)time queries for testing if a line lies above all the given lines. 4. An algorithm that tests the “towering property ” in O(n4/3+ε) time, for any ε>0: do n given red lines lie all above n given blue lines? The tools used to obtain these and other results include Plücker coordinates for lines in space and εnets for various geometric range spaces.
Cuttings and Applications
, 1993
"... We prove a general lemma on the existence of (1/r)cuttings of geometric objects in I = that stisfy certain properties. We use this lemma to construct (1/r)cuttings of (azymptotically) optimal size for arrangements of line segments in the plane and arrangements of triangles in 3space; for line ..."
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Cited by 18 (0 self)
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We prove a general lemma on the existence of (1/r)cuttings of geometric objects in I = that stisfy certain properties. We use this lemma to construct (1/r)cuttings of (azymptotically) optimal size for arrangements of line segments in the plane and arrangements of triangles in 3space; for line segments in the plane we obtain a cutting of size O(r + Ar2/n2), and for triangles in 3space our cutting haz size O(r2(r) + Ara/nZ). Here A is the combinatorial complexity of the arrangement. Finally, we use these results to obtain new results for several problems concerning line segments in the plane and triangles in 3space.
Metric combinatorics of convex polyhedra: Cut loci and nonoverlapping unfoldings
, 2003
"... Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by id ..."
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Cited by 14 (3 self)
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Abstract. Let S be the boundary of a convex polytope of dimension d + 1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into R d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v ∈ S, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3polytopes into R 2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic nonpolynomial complexity of nonconvex manifolds.
Computational Geometry
 in Directions in Computational Geometry
, 1994
"... Computational geometry, the study of algorithms involving relatively simple geometric objects, is an active, exciting field. This chapter samples current research in computational geometry. Three topics are discussed at some length: the theory of arrangements, a randomincremental convex hull alg ..."
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Cited by 10 (0 self)
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Computational geometry, the study of algorithms involving relatively simple geometric objects, is an active, exciting field. This chapter samples current research in computational geometry. Three topics are discussed at some length: the theory of arrangements, a randomincremental convex hull algorithm, and robustness of geometric algorithms. 1 Introduction As pointed out by Robin Forrest and others[39, 79], the term "computational geometry" could be used for a variety of fields, including geometric modeling using curves and surfaces, computer proofs of geometric theorems, geometric design software, and the theory of perceptrons[71]. The purpose of this chapter is to give an impressionistic view of recent research in computational geometry, defined as the study of algorithms involving relatively simple geometric objects such as points and lines. A typical topic in this field is the analysis of an algorithm involving n objects in d dimensions; examples are the computation of the c...
Counting Circular Arc Intersections
, 2002
"... In this paper we present efficient algorithms for counting intersections in a collection of circles or circular arcs. We present an algorithm to count intersections in a collection of n circles whose running time is O(n ), for any > 0. Using this algorithm as a subroutine, we show that the intersec ..."
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Cited by 10 (2 self)
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In this paper we present efficient algorithms for counting intersections in a collection of circles or circular arcs. We present an algorithm to count intersections in a collection of n circles whose running time is O(n ), for any > 0. Using this algorithm as a subroutine, we show that the intersections in a set of n circular arcs can also be counted in time O(n ). If all arcs have the same radius, the running time can be improved to O(n ), for any > 0.
An Efficient MultiDimensional Searching Technique and its Applications
, 1993
"... This paper describes an improved algorithm for the multidimensional searching problem introduced by Megiddo. As a result, we obtain a d O(d) n time deterministic algorithms for linear programming in R d with n constraints, for computing the Euclidean 1center of a set of n points in R d , for ..."
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Cited by 10 (3 self)
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This paper describes an improved algorithm for the multidimensional searching problem introduced by Megiddo. As a result, we obtain a d O(d) n time deterministic algorithms for linear programming in R d with n constraints, for computing the Euclidean 1center of a set of n points in R d , for computing the minimum enclosing ellipsoid of a set of n points in R d , etc. Our techniques also improve the running time of known algorithms for a number of parametric graph searching problems, including that of finding zero cycles in dynamic graphs [7].