Voronoi diagrams can also be thought of as lower envelopes, in the sense mentioned at the beginning of this subsection. Namely, for each point x not situated on a bisecting curve, the relation p x q defines a total ordering on S. If we construct a set of surfaces H p , p S,in3-space such that H p is below H q i# p x q holds, then the projection of their lower envelope equals the abstract Voronoi diagram.

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...

A convex polytope P can be speci ed in two ways: as the convex hull of the vertex set V of P, or as the intersection of the set H of its facet-inducing halfspaces. The vertex enumeration problem is to compute V from H. The facet enumeration problem it to compute H from V. These two problems are essentially equivalent under point/hyperplane duality. They are among the central computational problems in the theory of polytopes. It is open whether they can be solved in time polynomial in jHj + jVj. In this paper we consider the main known classes of algorithms for solving these problems. We argue that they all have at least one of two weaknesses: inability todealwell with "degeneracies," or, inability tocontrol the sizes of intermediate results. We then introduce families of polytopes that exercise those weaknesses. Roughly speaking, fat-lattice or intricate polytopes cause algorithms with bad degeneracy handling to perform badly; dwarfed polytopes cause algorithms with bad intermediate size control to perform badly. We also present computational experience with trying to solve these problem on these hard polytopes, using various implementations of the main algorithms.

We introduce the concept of a sensitive E-approx-imation, and use it to derive a more efficient algorithm for computing &-nets. We define and investigate prod-uct range spaces, for which we establish sampling the-orems analogous to the standard finite VC-dimensional case. This generalizes and simplifies results from previ-ous works. We derive a simpler optimal deterministic convex hull algorithm, and by extending the method to the intersection of a set of balls with the same radius, we obtain an O(n log3 n) deterministic algorithm for com-puting the diameter of an n-point set in 3-dimensional space.

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.

The Voronoi diagram of a point set is a fundamental geometric structure that partitions the space into elementary regions of influence defining a discrete proximity graph and dually a well-shaped Delaunay triangulation. In this paper, we investigate a framework for defining and building the Voronoi diagrams for a broad class of distortion measures called Bregman divergences, that includes not only the traditional (squared) Euclidean distance, but also various divergence measures based on entropic functions. As a by-product, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We show that for a given Bregman divergence, one can define several types of Voronoi diagrams related to each other

Given n points in three dimensions, we show how to answer halfspace range reporting queries in O(logn+k) expected time for an output size k. Our data structure can be preprocessed in optimal O(n log n) expected time. We apply this result to obtain the first optimal randomized algorithm for the construction of the ( k)-level in an arrangement of n planes in three dimensions. The algorithm runs in O(n log n+nk²) expected time. Our techniques are based on random sampling. Applications in two dimensions include an improved data structure for "k nearest neighbors" queries, and an algorithm that constructs the order-k Voronoi diagram in O(n log n + nk log k) expected time.

We show that in the worst case, Ω(n dd=2e\Gamma1 +n log n) sidedness queries are required to determine whether the convex hull of n points in R^d is simplicial, or to determine the number of convex hull facets. This lower bound matches known upper bounds in any odd dimension. Our result follows from a straightforward adversary argument. A key step in the proof is the construction of a quasi-simplicial n-vertex polytope with Ω(n dd=2e\Gamma1 ) degenerate facets. While it has been known for several years that d-dimensional convex hulls can have Ω(n bd=2c ) facets, the previously best lower bound for these problems is only Ω(n log n). Using similar techniques, we also obtain simple and correct proofs of Erickson and Seidel's lower bounds for detecting affine degeneracies in arbitrary dimensions and circular degeneracies in the plane. As a related result, we show that detecting simplicial convex hulls in R^d is ⌈d/2⌉-hard, in the in the sense of Gajentaan and Overmars.

We show that the well-known random incremental construction of Clarkson and Shor [14] can be adapted via gradations to provide efficient external-memory algorithms for some geometric problems. In particular, as the main result, we obtain an optimal randomized algorithm for the problem of computing the trapezoidal decomposition determined by a set of N line segments in the plane with K pairwise intersections, that requires \Theta( log M=B N ) expected disk accesses, where M is the size of the available internal memory and B is the size of the block transfer. The approach is sufficiently general to obtain algorithms also for the problems of 3-d half-space intersections, 2-d and 3-d convex hulls, 2-d abstract Voronoi diagrams and batched planar point location, which require an optimal expected number of disk accesses and are simpler than the ones previously known. The results extend to an external-memory model with multiple disks. Additionally, under reasonable conditions on the parameters N;M;B, these results can be notably simplified originating practical algorithms which still achieve optimal expected bounds.

Abstract. Computing reachable sets is an essential step in most analysis and synthesis techniques for hybrid systems. The representation of these sets has a deciding impact on the computational complexity and thus the applicability of these techniques. This paper presents a new approach for approximating reachable sets using oriented rectangular hulls (ORHs), the orientations of which are determined by singular value decompositions of sample covariance matrices for sets of reachable states. The orientations keep the over-approximation of the reachable sets small in most cases with a complexity of low polynomial order with respect to the dimension of the continuous state space. We show how the use of ORHs can improve the efficiency of reachable set computation significantly for hybrid systems with nonlinear continuous dynamics.