... process a set S of points in so that the points of S lying inside a query R region can be reported or counted quickly. Wesurvey the known techniques and data structures for range searching and describe their application to other related searching problems.

We review the recent progress in the design of efficient algorithms for various problems in geometric optimization. We present several techniques used to attack these problems, such as parametric searching, geometric alternatives to parametric searching, prune-and-search techniques for linear programming and related problems, and LPtype problems and their efficient solution. We then describe a variety of applications of these and other techniques to numerous problems in geometric optimization, including facility location, proximity problems, statistical estimators and metrology, placement and intersection of polygons and polyhedra, and ray shooting and other query-type problems.

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...

Range searching is one of the central problems in computational geometry, because it arises in many applications and a wide variety of geometric problems can be formulated as a range-searching problem. A typical range-searching problem has the following form. Let S be a set of n points in R d , and let R be a family of subsets; elements of R are called ranges . We wish to preprocess S into a data structure so that for a query range R, the points in S " R can be reported or counted efficiently. Typical examples of ranges include rectangles, halfspaces, simplices, and balls. If we are only interested in answering a single query, it can be done in linear time, using linear space, by simply checking for each point p 2 S whether p lies in the query range.

We present an accurate and efficient algorithm to compute the internal Voronoi region and medial axis of a 3-D polyhedron. It uses exact arithmetic and representations for accurate computation of the medial axis. The sheets, seams, and junctions of the medial axis are represented as trimmed quadric surfaces, algebraic space curves, and algebraic numbers, respectively. The algorithm works by recursively finding neighboring junctions along the axis. It utilizes discretization of space and linear programming to speed up the search step. We also present a new algorithm for analysis of the topology of an algebraic plane curve, which is the core of our medial axis algorithm. To speed up the computation, we have designed specialized algorithms for fast computation on implicit geometric structures. These include lazy evaluation based on multivariate Sturm sequences, fast resultant computation, curve topology analysis, and floating-point filters. The algorithm has been implemented and we high...

We give efficient algorithms for solving several geometric problems in computational metrology, focusing on the fundamental issues of "flatness" and "roundness." Specifically, we give approximate and exact algorithms for 2- and 3-dimensional roundness primitives, deriving results that improve previous approaches in several respects, including problem definition, running time, underlying computational model, and dimensionality of the input. We also study methods for determining the width of a d-dimensional point set, which corresponds to the metrology notion of "flatness," giving an approximation method that can serve as a fast exactcomputation filter for this metrology primitive. Finally, we report on experimental results derived from implementation and testing, particularly in 3-space, of our approximation algorithms, including several heuristics designed to significantly speed-up the computations in practice.

We show that the complexity of the vertical decomposition of an arrangement of n fixeddegree algebraic surfaces or surface patches in four dimensions is O(n 4+ε), for any ε> 0. This improves the best previously known upper bound for this problem by a near-linear factor, and settles a major problem in the theory of arrangements of surfaces, open since 1989. The new bound can be extended to higher dimensions, yielding the bound O(n 2d−4+ε), for any ε> 0, on the complexity of vertical decompositions in dimensions d ≥ 4. We also describe the immediate algorithmic applications of these results, which include improved algorithms for point location, range searching, ray shooting, robot motion planning, and some geometric optimization problems. 1

We describe the relevance of Computational Geometry to tolerancing metrology. We outline the basic issues and define the class of zone problems that is central in this area. In the context of the exact computation paradigm, these problems are prime candidates for "exact solution" since we show that they have bounded-depth. Metrologists in this field have mounted a quest for a reference software which will impose some certainty in a confusing market of metrology software. The use of exact computation in the reference software will solve many intractable difficulties associated with current approaches. In short, here is a practical area in which CG and exact computation can have a real impact. 1 Introduction Researchers in Computational Geometry (CG) have always been convinced that their subject is relevant to a variety of application areas. But CG'ers have often assumed that the application areas would come to CG to find answers to their questions. To what extent is this valid? I will d...

INTRODUCTION Given a finite collection S of geometric objects such as hyperplanes or spheres in R d , the arrangement A(S) is the decomposition of R d into connected open cells of dimensions 0; 1; : : :; d induced by S. Besides being interesting in their own right, arrangements of hyperplanes have served as a unifying structure for many problems in discrete and computational geometry. With the recent advances in the study of arrangements of curved (algebraic) surfaces, arrangements have emerged as the underlying structure of geometric problems in a variety of `physical world' application domains such as robot motion planning and computer vision. This chapter is devoted to arrangements of hyperplanes and of curved surfaces in low-dimensional Euclidean space, with an emphasis on combinatorics and algorithms. In the first section we in

this paper addresses and survey previous work on these problems. We state the basic new results in Section 3. We exemplify the usefulness of these results by applying them to problems involving robot motion planning (Section 4) and visibility and aspect graphs (Section 5). Section 6 deals with new results on Minkowski sums of convex polyhedra in three dimensions, which have applications in robot motion planning and in other related areas. The paper concludes in Section 7, with further applications of the new results and with some open problems.