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 range searching, algorithmic problems of the following type are considered: Given an n-point set P in the plane, build a data structure so that, given a query triangle R, the number of points of P lying in R can be determined quickly. Problems of this type are of crucial importance in computational geometry, as they can be used as subroutines in many seemingly unrelated algorithms. We present a survey of results and main techniques in this area.

This paper describes new methods for maintaining a point-location data structure for a dynamically-changing monotone subdivision S. The main approach is based on the maintenance of two interlaced spanning trees, one for S and one for the graphtheoretic planar dual of S. Queries are answered by using a centroid decomposition of the dual tree to drive searches in the primal tree. These trees are maintained via the link-cut trees structure of Sleator and Tarjan, leading to a scheme that achieves vertex insertion/deletion in O(log n) time, insertion/deletion of k-edge monotone chains in O(log n + k) time, and answers queries in O(log 2 n) time, with O(n) space, where n is the current size of subdivision S. The techniques described also allow for the dual operations expand and contract to be implemented in O(log n) time, leading to an improved method for spatial point-location in a 3-dimensional convex subdivision. In addition, the interlaced-tree approach is applied to on-line point-lo...

Abstract. Let F be a collection of nd-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(n d+ε) for any ε>0. For d = 3, by combining this algorithm with the point-location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n 3+ε), for any ε>0, a data structure of size O(n 3+ε) that, for any query point q, can determine in O(log 2 n) time the function(s) of F that attain the lower envelope at q. As a consequence, we obtain improved algorithmic solutions to several problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the “biggest stick ” in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in randomized expected time O(n 17/11+ε), for any ε>0, improving previous solutions that run in time O(n 8/5+ε). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n 3+ε) storage and preprocessing time, for any ε>0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

We define two results on approximate shortest path maps in IR 3 . (i) Given a polyhedral surface or a convex polytope P with n edges in IR 3 , a source point s on P , and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision of P of size O((n=") log(1=")) which can be used to answer efficiently approximate shortest path queries. Namely, given any point t on P , one can compute, in O(log (n=")) time, a distance \Delta P;s (t), such that dP;s (t) \Delta P;s (t) (1 + ")d P;s (t), where dP;s (t) is the length of a shortest path between s and t on P . The map can be computed in O(n 2 log n + (n=") log (1=") log (n=")) time, for the case of a polyhedral surface, and in O((n=" 3 ) log(1=") + (n=" 1:5 ) log (1=") log n) time if P is a convex polytope. (ii) Given a set of polyhedral obstacles O with a total of n edges in IR 3 , a source point s in IR 3 n int S O2O O, and a real parameter 0 ! " 1, we present an algorithm that computes a subdivision o...

Given a polygonal object (simple polygon, geometric graph, wire-frame, skeleton or more generally a set of line segments) in three dimensional Euclidean space, we consider the problem of computing a variety of "nice" parallel (orthographic) projections of the object. We show that given a general polygonal object consisting of n line segments in space, deciding whether it admits a crossing-free projection can be done in O(n 2 log n+k) time and O(n 2 +k) space, where k is the number of edge intersections of forbidden quadrilaterals (i.e. set of directions that admits a crossing) and varies from zero to O(n 4 ). This implies for example that given a simple polygon in 3-space we can determine if there exists a plane on which the projection is a simple polygon, within the same complexity. Furthermore, if such a projection does not exist, a minimum-crossing projection can be found in O(n 4 ) time and space. We show that an object always admits a regular projection (of interest to k...

) To appear in Proc. Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '99), January 17-19, 1999 Yi-Jen Chiang Joseph S. B. Mitchell y Abstract We consider the two-point query version of the fundamental geometric shortest path problem: Given a set h of polygonal obstacles in the plane, having a total of n vertices, build a data structure such that for any two query points s and t we can efficiently determine the length, d(s; t), of an Euclidean shortest obstacle-avoiding path, ß(s; t), from s to t. Additionally, our data structure should allow one to report the path ß(s; t), in time proportional to its (combinatorial) size. We present various methods for solving this two-point query problem, including algorithms with o(n), O(log n+h), O(h log n), O(log 2 n) or optimal O(log n) query times, using polynomial-space data structures, with various tradeoffs between space and query time. While several results have been known for approximate two-point Euclidean shortest p...

this paper, except Figures 4.8 and 4.9, are two dimensional, representing polyhedra as polygons. 1 2 3 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 4 * vp Figure 4.1: Visibility ordering of the cells of a mesh relative to viewpoint vp. can be computed and stored in a preprocessing step. The MPVO algorithm can be extended to order many nonconvex meshes; this is described in detail in Section 4.4.2. 4.3 Preliminary Definitions A convex polyhedron in E

We study the motion-planning problem for a convex m-gon P in a planar polygonal environment Q bounded by n edges. We give the first algorithm that constructs the entire free configuration space (the 3-dimensional space of all free placements of P in Q) in time that is near-quadratic in mn, which is nearly optimal in the worst case. The algorithm is also conceptually relatively simple. Previous solutions were incomplete, more expensive, or produced only part of the free configuration space. Combining our solution with parametric searching, we obtain an algorithm that finds the largest placement of P in Q in time that is also near-quadratic in mn. In addition, we describe an algorithm that preprocesses the computed free configuration space so that `reachability' queries can be answered in polylogarithmic time. All three authors have been supported by a grant from the U.S.-Israeli Binational Science Foundation. Pankaj Agarwal has also been supported by a National Science Foundation Gr...

In the context of methodologies intended to confer robustness to geometric algorithms, we elaborate on the exact computation paradigm and formalize the notion of degree of a geometric algorithm, as a worst-case quantification of the precision (number of bits) to which arithmetic calculation have to be executed in order to guarantee topological correctness. We also propose a formalism for the expeditious evaluation of algorithmic degree. As an application of this paradigm and an illustration of our general approach, we consider the important classical problem of proximity queries in 2 and 3 dimensions, and develop a new technique for the efficient and robust execution of such queries based on an implicit representation of Voronoi diagrams. Our new technique gives both low degree and fast query time, and for 2D queries is optimal with respect to both cost measures of the paradigm, asymptotic number of operations and arithmetic degree.