The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH04-96-1-0013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.-Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR-91-22103 and CCR-93-11127, by a Max-Planck Research Award, and by grants from the U.S.-Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...

In geometric pattern matching, we are given two sets of points P and Q in d dimensions, and the problem is to determine the rigid transformation that brings P closest to Q, under some distance measure. More generally, each point can be modelled as a ball of small radius, and we may wish to nd a transformation approximating the closest distance between P and Q. This problem has many applications in domains such as computer vision and computational chemistry In this paper we present improved algorithms for this problem, by allowing the running time of our algorithms to depend not only on n, (the number of points in the sets), but also on, the diameter of the point set. The dependence on also allows us to e ectively process point sets that occur in practice, where diameters tend to be small ([EVW94]). Our algorithms are also simple to implement, in contrast to much of the earlier work. To obtain the above-mentioned results, we introduce a novel discretization technique to reduce geometric pattern matching to combinatorial pattern matching. In addition, we address various generalizations of the classical problem rst posed by Erdos: \Given a set of n points in the plane, how many pairs of points can be exactly a unit distance apart?". The combinatorial bounds we prove enable us to obtain improved results for geometric pattern matching and may have other applications.

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in R d , is any point contained in any hyperplane? We define a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time #(n log m+n 2/3 m 2/3 +m log n) in two dimensions, or #(n log m+n 5/6 m 1/2 +n 1/2 m 5/6 + m log n) in three or more dimensions. We obtain slightly higher bounds for the counting version of Hopcroft's problem in four or more dimensions. Our planar lower bound is within a factor of 2 O(log # (n+m)) of the best known upper bound, due to Matousek. Previously, the best known lower bound, in any dimension, was #(n log m + m log n). We develop our lower bounds in two stages. First we define a combinatorial representation of the relative order type of a set of points and hyperplanes, called a monochromatic cover, and derive low...

A geometric hypergraph H is a collection of i-dimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)-tuples of a vertex set V in general position in d-space. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the k-set problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (i-simplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges...

We consider the relative complexities of a large number of computational geometry problems whose complexities are believed to be roughly \Theta(n 4=3 ). For certain pairs of problems, we show that the complexity of one problem is asymptotically bounded by the complexity of the other. Almost all of the problems we consider can be solved in time O(n 4=3+ffi ) or better, and there are\Omega\Gamma n 4=3 ) lower bounds for a few of them in specialized models of computation. However, the best known lower bound in any general model of computation is only\Omega\Gamma n log n). The paper is naturally divided into two parts. In the first part, we consider a large number of problems that are harder than Hopcroft's problem. These problems include various ray shooting problems, sorting line segments in IR 3 , collision detection in IR 3 , and halfspace emptiness checking in IR 5 . In the second, we survey known reductions among problems involving lines in three-space, and among highe...

this paper, we only consider

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there

We develop the first nontrivial lower bounds on the complexity of online hyperplane and halfspace emptiness queries. Our lower bounds apply to a general class...

We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.

A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turán-type and Ramsey-type extremal problems for geometric graphs, and discusses their generalizations and applications.