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20
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.
Raising Roofs, Crashing Cycles, and Playing Pool: Applications of a Data Structure for Finding Pairwise Interactions
 In Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an ngon with r reflex vertices in time O(n 1+" +n 8=11+" r ..."
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Cited by 46 (1 self)
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The straight skeleton of a polygon is a variant of the medial axis, introduced by Aichholzer et al., defined by a shrinking process in which each edge of the polygon moves inward at a fixed rate. We construct the straight skeleton of an ngon with r reflex vertices in time O(n 1+" +n 8=11+" r 9=11+" ), for any fixed " ? 0, improving the previous best upper bound of O(nr log n). Our algorithm simulates the sequence of collisions between edges and vertices during the shrinking process, using a technique of Eppstein for maintaining extrema of binary functions to reduce the problem of finding successive interactions to two dynamic range query problems: (1) maintain a changing set of triangles in IR 3 and answer queries asking which triangle would be first hit by a query ray, and (2) maintain a changing set of rays in IR 3 and answer queries asking for the lowest intersection of any ray with a query triangle. We also exploit a novel characterization of the straight skeleton as a ...
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 35 (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...
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.
Tentative PruneandSearch for Computing FixedPoints with Applications to Geometric Computation
 Fundamenta Informaticae
, 1995
"... . Motivated by problems in computational geometry, this paper investigates the complexity of finding a fixedpoint of the composition of two or three continuous functions that are defined piecewise. It shows that certain cases require nested binary search taking \Theta(log 2 n) time. Other cases ..."
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Cited by 18 (4 self)
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. Motivated by problems in computational geometry, this paper investigates the complexity of finding a fixedpoint of the composition of two or three continuous functions that are defined piecewise. It shows that certain cases require nested binary search taking \Theta(log 2 n) time. Other cases can be solved in logarithmic time by using a pruneand search technique that may make tentative discards and later revoke or certify them. This work finds application in optimal subroutines that compute approximations to convex polygons, dense packings, and Voronoi vertices for Euclidean and polygonal distance functions. 1 Introduction Several fundamental problems in computational geometry can be expressed as a search for special ktuple with one element drawn from each of k lists. Examples with k equal to 2 or 3 include common tangents to two convex polygons, special chords in a polygon, and Voronoi vertices (circles or polygons tangent to three given points or polygons). Many of these p...
The predicates for the Voronoi diagram of ellipses
 In Proc. 22th Annual ACM Symp. on Computational Geometry
, 2006
"... This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exter ..."
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Cited by 13 (8 self)
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This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to nonintersecting ellipses, but the extension to arbitrary ones is straightforward. The ellipses are input in parametric representation or constructively. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivisionbased algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.
Faster Circle Packing with Application to Nonobtuse Triangulation
, 1994
"... We show how to pack a nonsimple polygon with O(n) tangent circles, so that each remaining region is adjacent to at most four circles, in total time O(n log n). This improves a previous O(n log 2 n) bound. As a consequence, we can triangulate the polygon with right and acute triangles in the s ..."
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Cited by 10 (3 self)
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We show how to pack a nonsimple polygon with O(n) tangent circles, so that each remaining region is adjacent to at most four circles, in total time O(n log n). This improves a previous O(n log 2 n) bound. As a consequence, we can triangulate the polygon with right and acute triangles in the same bounds. # Work supported in part by NSF grant CCR9258355. Figure 1. Nonobtuse triangulation steps: (a) pack polygon with circles; (b) connect circle centers; (c) triangulate remaining polygonal regions. (Courtesy of M. Bern) 1 Introduction Bern et al. [2] recently described a new algorithm for triangulating any polygonal region (possibly with holes), in such a way that no triangle has an acute angle; such triangulations are useful as unstructured meshes for the finite element method. Their method only adds O(n) new Steiner points, improving previous quadratic methods [1]. However the algorithm they describe takes O(n log 2 n) time. In this paper we describe a technique for perfor...
Queries with Segments in Voronoi Diagrams
, 1999
"... In this paper we consider proximity problems in which the queries are line segments in the plane. We build a query structure that for a set of n points P can determine the closest point in P to a query segment outside the convex hull of P in O(log n) time. With this we solve the problem of computing ..."
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Cited by 10 (1 self)
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In this paper we consider proximity problems in which the queries are line segments in the plane. We build a query structure that for a set of n points P can determine the closest point in P to a query segment outside the convex hull of P in O(log n) time. With this we solve the problem of computing the closest point to each of n disjoint line segments in O(n log 3 n) time. Nearest foreign neighbors or Hausdorff distance for disjoint, colored segments can be computed in the same time. We explore some connections to Hopcroft's problem. 1 Introduction Since Knuth [13] posed the post office problem preprocess a set of points, or sites, in the plane to quickly report the nearest to a query pointand Shamos and Hoey [17] suggested Voronoi diagrams as a solution, there have been a number of proximity problems in the plane whose solution is to build some type of Voronoi diagram and query with a point. Note: A Voronoi diagram of a set of sites is the partition of the plane into maxim...
Compact Voronoi Diagrams for Moving Convex Polygons
 In Proc. Scand. Workshop Alg. and Data Structures (SWAT
, 2000
"... We describe a kinetic data structure for maintaining a compact Voronoilike diagram of convex polygons moving around in the plane. We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96]. A key feature of this diagram is that its size is only a function of the n ..."
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Cited by 6 (4 self)
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We describe a kinetic data structure for maintaining a compact Voronoilike diagram of convex polygons moving around in the plane. We use a compact diagram for the polygons, dual to the Voronoi, first presented in [MKS96]. A key feature of this diagram is that its size is only a function of the number of polygons and not of their complexity. We demonstrate a local certifying property of that diagram, akin to that of Delaunay triangulations of points. We then obtain a method for maintaining this diagram that is outputsensitive and costs O(log n) per update. Furthermore, we show that for a set of k polygons with a total of n vertices moving along bounded degree algebraic motions, this dual diagram, and thus their compact Voronoi diagram, changes combinatorially## ) and O(kn #(k)#(n)) times, where #() is an extremely slowly growing function. This compact Voronoi diagram can be used for collision detection or retraction motion planning among the moving polygons.