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...

This paper addresses the problem of decomposing a complex polyhedral surface into a small number of "convex" patches (ie, boundary parts of convex polyhedra). The corresponding optimization problem is shown to be NP-complete and an experimental search for good heuristics is undertaken. 1 Introduction Convex shapes are easiest to represent, manipulate, and render. Even though they form the building blocks of bottom-up solid modelers, it is more often the case that the convex structure of a geometric shape is lost in its representation. We are then presented, not with the solidmodeling problem of putting together primitive convex objects, but with the reverse problem of extracting convexity out of a complex shape. The classical example is that of cutting up a 3-polyhedron into convex pieces. This is often a useful, sometimes a required, preprocessing step in graphics, manufacturing, and mesh generation. The problem has been exhaustively researched in the last few years [2]---[18]. Despi...

This paper describes a method for estimating the distance between a robot and its surrounding environment using best ellipsoid fit. The method consists of the following two stages. First we approximate the detailed geometry of the robot and its environment by minimum-volume enclosing ellipsoids. The computation of these ellipsoids is a convex optimization problem, for which efficient algorithms are known. Then we compute a conservative distance estimate using an important but little known formula for the distance of a point from and n-dimensional ellipse. The computation of the distance estimate (and its gradient vector) is shown to be an eigenvalue problem, whose solution can be rapidly found using standard techniques. We also present an incremental version of the distance computation, which takes place along a continuous trajectory taken by the robot. We have implemented the proposed approach and present some preliminary results.

SUE WHITESIDES∗ ∗ Abstract. Motivated by visibility problems in three dimensions, we investigate the complexity and construction of the set of tangent lines in a scene of three-dimensional polyhedra. We prove that the set of lines tangent to four possibly intersecting convex polyhedra in R 3 with a total of n edges consists of Θ(n 2) connected components in the worst case. In the generic case, each connected component is a single line, but our result still holds for arbitrarily degenerate scenes. More generally, we show that a set of k possibly intersecting convex polyhedra with a total of n edges admits, in the worst case, Θ(n 2 k 2) connected components of maximal free line segments tangent to at least four polytopes. Furthermore, these bounds also hold for possibly occluded lines rather than maximal free line segments. Finally, we present an O(n 2 k 2 log n) time and O(nk 2) space algorithm that, given a scene of k possibly intersecting convex polyhedra, computes all the minimal free line segments that are tangent to any four of the polytopes and are isolated transversals to the set of edges they intersect; in particular, we compute at least one line segment per connected component of tangent lines. Key words. computational geometry, 3D visibility, visibility complex, visual events

. Let B be a convex polyhedron translating in 3-space amidst k convex polyhedral obstacles A 1 , . . . , A k with pairwise disjoint interiors. The free configuration space (space of all collision-free placements) of B can be represented as the complement of the union of the Minkowski sums P i = A i # (-B), for i = 1, . . . , k. We show that the combinatorial complexity of the free configuration space of B is O(nk log k), and that it can be ## nk#(k)) in the worst case, where n is the total complexity of the individual Minkowski sums P 1 , . . . , P k . We also derive an e#cient randomized algorithm that constructs this configuration space in expected time O(nk log k log n). Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms, algorithmic motion planning AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539794266602 1. Introduction. Let A 1 , . . . , A k be k close...

The ability to perform efficient collision detection is essential in virtual reality environments and their applications, such as walkthroughs. In this paper we re-explore a classical structure used for collision detection -- the binary space partitioning tree. Unlike the common approach, which attributes equal likelihood to each possible query, we assume events that happened in the past are more likely to happen again in the future. This leads us to the definition of self-customized data structures. We report encouraging results obtained while experimenting with this concept in the context of self-customized bsp trees. Keywords: Collision detection, binary space partitioning, self-customization. 1 Introduction Virtual reality refers to the use of computer graphics to simulate physical worlds or to generate synthetic ones, where a user is to feel immersed in the environment to the extent that the user feels as if "objects" seen are really there. For example, "objects" should m...

Given a polyhedral terrain T with n vertices, the two-watchtower problem for T asks to find two vertical segments, called watchtowers, of smallest common height, whose bottom endpoints (bases) lie on T, and whose top endpoints guard T, in the sense that each point on T is visible from at least one of them. There are three versions of the problem, discrete, semi-discrete, and continuous, depending on whether two, one, or none of the two bases are restricted to be among the vertices of T, respectively. In this paper we present the following results for the two-watchtower problem in R 2 and R 3: (1) We show that the discrete two-watchtowers problem in R 2 can be solved in O(n 2 log 4 n) time, significantly improving previous solutions. The algorithm works, without increasing its asymptotic running time, for the semi-continuous version, where one of the towers is allowed to be placed anywhere on T. (2) We show that the continuous two-watchtower problem in R 2 can be solved in O(n 3 α(n)log 3 n) time, again significantly improving previous results. (3) Still in R 2, we show that the continuous version of the problem of guarding a finite set P ⊂ T of m points by two watchtowers of smallest common height can be solved in O(mnlog 4 n) time.

We describe an algorithm for computing the intersection of n balls of equal radius in R³ which runs in time O(n lg² n). The algorithm can be parallelized so that the comparisons that involve the radius of the balls are performed in O(lg³ n) batches. Using parametric search, these algorithms are used to obtain an algorithm for computing the diameter of a set of n points in R³ (the maximum distance between any pair) which runs in time O(n lg^5 n). The algorithms are deterministic and elementary; this is in contrast with the running time O(n log n) in both cases that can be achieved using randomization [3], and the running times O(n lg n) and O(n lg³ n) using deterministic geometric sampling [2, 1].

We show how to use a combination of tree-clustering techniques and computational geometry to improve the time bounds for optimal pivot selection in the primal network simplex algorithm for minimum-cost flow and related problems and for pivot execution in the dual network simplex algorithm, from O(m) to O (√m) per pivot. Our techniques can also speed up network simplex algorithms for generalized flow, shortest paths with negative edges, maximum flow, the assignment problem, and the transshipment problem.