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58
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.
Approximating Polygons and Subdivisions with MinimumLink Paths
, 1991
"... We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate object ..."
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Cited by 61 (11 self)
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We study several variations on one basic approach to the task of simplifying a plane polygon or subdivision: Fatten the given object and construct an approximation inside the fattened region. We investigate fattening by convolving the segments or vertices with disks and attempt to approximate objects with the minimum number of line segments, or with near the minimum, by using efficient greedy algorithms. We give some variants that have linear or O(n log n) algorithms approximating polygonal chains of n segments. We also show that approximating subdivisions and approximating with chains with no selfintersections are NPhard.
Tree Methods for Moving Interfaces
, 1999
"... Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semiLagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effor ..."
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Cited by 56 (7 self)
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Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semiLagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effort on the interface, so the methods moves an interface with N degrees of freedom in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. The methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging, anisotropy, faceting and curvature.
Algorithmic Motion Planning
, 1997
"... INTRODUCTION Motion planning is a fundamental problem in robotics. It comes in a variety of forms, but the simplest version is as follows. We are given a robot system B, which may consist of several rigid objects attached to each other through various joints, hinges, and links, or moving independen ..."
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Cited by 43 (6 self)
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INTRODUCTION Motion planning is a fundamental problem in robotics. It comes in a variety of forms, but the simplest version is as follows. We are given a robot system B, which may consist of several rigid objects attached to each other through various joints, hinges, and links, or moving independently, and a twodimensional or threedimensional environment V cluttered with obstacles. We assume that the shape and location of the obstacles and the shape of B are known to the planning system. Given an initial placement Z 1 and a nal placement Z 2 of B, we wish to determine whether there exists a collisionavoiding motion of B from Z 1 to Z 2 , and, if so, to plan such a motion. In this simpli ed and purely geometric setup, we ignore issues such as incomplete information, nonholonomic constraints, control issues related to inaccuracies in sensing and motion, nonstationary obstacles, optimality of the planned motion, and so on. Since the early 1980's, motion planning has been an intensiv
Fast Treebased Redistancing for Level Set Computations
, 1999
"... Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents ecient algorithms for this "redistancing" problem. The algorith ..."
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Cited by 37 (6 self)
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Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents ecient algorithms for this "redistancing" problem. The algorithms use quadtrees and triangulation to compute global approximate signed distance functions. A quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants. Given a polygonal interface with N elements, our algorithms run in O(N) space and O(N log N) time. Twodimensional numerical results show they are highly efficient in practice.
Straight Skeletons for General Polygonal Figures in the Plane
, 1996
"... : A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable to th ..."
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Cited by 36 (1 self)
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: A novel type of skeleton for general polygonal figures, the straight skeleton S(G) of a planar straight line graph G, is introduced and discussed. Exact bounds on the size of S(G) are derived. The straight line structure of S(G) and its lower combinatorial complexity may make S(G) preferable to the widely used Voronoi diagram (or medial axis) of G in several applications. We explain why S(G) has no Voronoi diagram based interpretation and why standard construction techniques fail to work. A simple O(n) space algorithm for constructing S(G) is proposed. The worstcase running time is O(n 3 log n), but the algorithm can be expected to be practically efficient, and it is easy to implement. We also show that the concept of S(G) is flexible enough to allow an individual weighting of the edges and vertices of G, without changes in the maximal size of S(G), or in the method of construction. Apart from offering an alternative to Voronoitype skeletons, these generalizations of S(G) have ap...
Cut locus and medial axis in global shape interrogation and representation
 MIT Design Laboratory Memorandum 922 and MIT Sea Grant Report
, 1992
"... The cut locus CA of a closed set A in the Euclidean space E is defined as the closure of the set containing all points p which have at least two shortest paths to A. We present a theorem stating that the complement of the cut locus i.e. E\(CA∪A) is the maximal open set in (E\A) where the distance fu ..."
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Cited by 35 (1 self)
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The cut locus CA of a closed set A in the Euclidean space E is defined as the closure of the set containing all points p which have at least two shortest paths to A. We present a theorem stating that the complement of the cut locus i.e. E\(CA∪A) is the maximal open set in (E\A) where the distance function with respect to the set A is continuously differentiable. This theorem includes also the result that this distance function has a locally Lipschitz continuous gradient on (E\A). The medial axis of a solid D in E is defined as the union of all centers of all maximal discs which fit in this domain. We assume in the medial axis case that D is closed and that the boundary ∂D of D is a topological (not necessarily connected) hypersurface of E. Under these assumptions we prove that the medial axis of D equals that part of the cut locus of ∂D which is contained in D. We prove that the medial axis has the same homotopy type as its reference solid if the solid’s boundary surface fulfills certain regularity requirements. We also show that the medial axis with its related distance function can be be used to reconstruct its reference solid. We prove that the cut locus of a solid’s boundary is nowhere dense in the Euclidean space if the solid’s boundary meets certain regularity requirements. We show that the cut locus concept offers a common frame work lucidly unifying different concepts such as Voronoi diagrams, medial axes and equidistantial point sets. In this context we prove that the equidistantial set of two disjoint
Shape representation using a generalized potential field model
 IEEE TPAMI
, 1997
"... Abstract—This paper is concerned with efficient derivation of the medial axis transform of a twodimensional polygonal region. Instead of using the shortest distance to the region border, a potential field model is used for computational efficiency. The region border is assumed to be charged and the ..."
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Cited by 31 (0 self)
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Abstract—This paper is concerned with efficient derivation of the medial axis transform of a twodimensional polygonal region. Instead of using the shortest distance to the region border, a potential field model is used for computational efficiency. The region border is assumed to be charged and the valleys of the resulting potential field are used to estimate the axes for the medial axis transform. The potential valleys are found by following force field, thus, avoiding twodimensional search. The potential field is computed in closed form using the equations of the border segments. The simple Newtonian potential is shown to be inadequate for this purpose. A higher order potential is defined which decays faster with distance than as inverse of distance. It is shown that as the potential order becomes arbitrarily large, the axes approach those computed using the shortest distance to the border. Algorithms are given for the computation of axes, which can run in linear parallel time for part of the axes having initial guesses. Experimental results are presented for a number of examples. Index Terms—Generalized potential, Newtonian potential, topology, medial axis, symmetric axis transform, skeletonization, distance
SemiLagrangian Methods for Level Set Equations
, 1998
"... A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semiLagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the ..."
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Cited by 31 (6 self)
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A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semiLagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the CFL stability condition, permitting the construction of methods which are efficient, adaptive and modular. Analysis of a linear onedimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of twodimensional numerical results. The new method computes correct viscosity solutions to problems involving geometry, anisotropy, curvature and complex topological events.
On Fat Partitioning, Fat Covering and the Union Size of Polygons
, 1993
"... The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has unio ..."
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Cited by 30 (2 self)
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The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.