Open AccessBook
Arrangements of Curves in the Plane- Topology, Combinatorics, and Algorithms
Herbert Edelsbrunner,Leonidas J. Guibas,János Pach,Richard Pollack,Raimund Seidel,Micha Sharir +5 more
TLDR
A generalization of the zone theorem of [EOS], [CGL] to arrangements of curves, and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.Abstract:
Arrangements of curves in the plane are of fundamental significance in many problems of computational and combinatorial geometry (e.g. motion planning, algebraic cell decomposition, etc.). In this paper we study various topological and combinatorial properties of such arrangements under some mild assumptions on the shape of the curves, and develop basic tools for the construction, manipulation, and analysis of these arrangements. Our main results include a generalization of the zone theorem of [EOS], [CGL] to arrangements of curves (in which we show that the combinatorial complexity of the zone of a curve is nearly linear in the number of curves), and an application of (some weaker variant of) that theorem to obtain a nearly quadratic incremental algorithm for the construction of such arrangements.read more
Citations
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Davenport-Schinzel sequences and their geometric applications
Micha Sharir,Pankaj K. Agarwal +1 more
TL;DR: A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
Journal ArticleDOI
Combinatorial complexity bounds for arrangements of curves and spheres
TL;DR: Upper and lower bounds for extremal problems defined for arrangements of lines, circles, spheres, and alike are presented and it is proved that the maximum number of edges boundingm cells in an arrangement ofn lines is Θ(m2/3n 2/3 +n), and that it isO(m3/2β(m) forn unit-circles.
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Arrangements and Their Applications
Pankaj K. Agarwal,Micha Sharir +1 more
TL;DR: This chapter surveys combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions and presents many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization.
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The upper envelope of voronoi surfaces and its applications
TL;DR: Borders on the number of vertices on the upper envelope of a collection of Voronoi surfaces are derived, and efficient algorithms to calculate these vertices are provided.
Journal ArticleDOI
Fat Triangles Determine Linearly Many Holes
TL;DR: The authors show that for every fixed $\delta>0$ the following holds: if $F$ is a union of triangles, all of whose angles are at least $delta$, then the complement of F has $O(n)$ connected components and the boundary of F consists of straight segments.
References
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Book
Algorithms in Combinatorial Geometry
TL;DR: This book offers a modern approach to computational geo- metry, an area thatstudies the computational complexity of geometric problems with an important role in this study.
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Primitives for the manipulation of general subdivisions and the computation of Voronoi
Leonidas J. Guibas,Jorge Stolfi +1 more
TL;DR: The following problem is discussed: given n points in the plane (the sites) and an arbitrary query point q, find the site that is closest to q, which can be solved by constructing the Voronoi diagram of the griven sites and then locating the query point in one of its regions.
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On the “piano movers'” problem I. The case of a two‐dimensional rigid polygonal body moving amidst polygonal barriers
Jacob T. Schwartz,Micha Sharir +1 more
TL;DR: In this paper, a two-dimensional case of the problem is solved, where given a body B and a region bounded by a collection of "walls", either find a continuous motion connecting two given positions and orientations of B during which B avoids collision with the walls, or else establish that no such motion exists.
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Constructing arrangements of lines and hyperplanes with applications
TL;DR: An algorithm is presented that constructs a representation for the cell complex defined by n hyperplanes in optimal $O(n^d )$ time in d dimensions, which is shown to lead to new methods for computing $\lambda $-matrices, constructing all higher-order Voronoi diagrams, halfspatial range estimation, degeneracy testing, and finding minimum measure simplices.
Journal ArticleDOI
On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles
TL;DR: An upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets is obtained and can be applied to planning a collision-free translational motion of a convex polygonB amidst several polygonal obstacles.