. We show that the number of vertices, edges, and faces of the union of k convex polyhedra in 3-space, having a total of n faces, is O(k 3 + kn log k). This bound is almost tight in the worst case, as there exist collections of polyhedra with## k 3 + kn#(k)) union complexity. We also describe a rather simple randomized incremental algorithm for computing the boundary of the union in O(k 3 + kn log k log n) expected time. Key words. combinatorial geometry, computational geometry, combinatorial complexity, convex polyhedra, geometric algorithms, randomized algorithms AMS subject classifications. 52B10, 52B55, 65Y25, 68Q25, 68U05 PII. S0097539793250755 1. Combinatorial bounds. Let P = {P 1 , . . . , P k } be a family of k convex polyhedra in 3-space, let n i be the number of faces of P i , and let n = # k i=1 n i . Put U = # P. By the combinatorial complexity of a polyhedral set we mean the total number of its vertices, edges, and faces. Our main result is the followin...

We introduce a new type of randomized incremental algorithms. Contrary to standard randomized incremental algorithms, these lazy randomized incremental algorithms are suited for computing structures that have a `non-local' definition. In order to analyze these algorithms we generalize some results on random sampling to such situations. We apply our techniques to obtain efficient algorithms for the computation of single cells in arrangements of segments in the plane, single cells in arrangements of triangles in space, and zones in arrangements of hyperplanes. We also prove combinatorial bounds on the complexity of what we call the (6k)-cell in arrangements of segments in the plane or triangles in space; this is the set of all points on the segments (triangles) that can reach the origin with a path that crosses at most k, 1 segments (triangles).

The complexity of the contour of the union of simple polygons with n vertices in total can be O(n 2) in general. A notion of fatness for simple polygons is introduced, which extends most of the existing fatness definitions. It is proved that a set of fat polygons with n vertices in total has union complexity is O(nloglogn), which is a generalization of a similar result for fat triangles [19]. Applications to several basic problems in computational geometry are given, such as efficient hidden surface removal, motion planning, injection molding, etc. The result is based on a new method to partition a fat simple polygon P with n vertices into O(n) fat convex quadrilaterals, and a method to cover (but not partition) a fat convex quadrilateral with O(1) fat triangles. The maximum overlap of the triangles at any point is two, which is optimal for any coveting of a fat simple polygon by a linear number of fat triangles.

We present a new data structure for a set of n convex simply-shaped fat objects in the plane, and use it to obtain efficient and rather simple solutions to several problems including (i) vertical ray shooting --- preprocess a set K of n non-intersecting convex simply-shaped flat objects in 3-space, whose xy-projections are fat, for efficient vertical ray shooting queries, (ii) point enclosure --- preprocess a set C of n convex simply-shaped fat objects in the plane, so that the k objects containing a query point p can be reported efficiently, (iii) bounded-size range searching --- preprocess a set C of n convex fat polygons, so that the k objects intersecting a `not-too-large' query polygon can be reported efficiently, and (iv) bounded-size segment shooting --- preprocess a set C as in (iii), so that the first object (if exists) hit by a `not-too-long' oriented query segment can be found efficiently. For the first three problems we construct data structures of size O(s (n) log 3 n)...

Let K be a set of n non-intersecting objects in 3-space. A depth order of K, if exists, is a linear order ! of the objects in K such that if K;L 2 K and K lies vertically below L then K ! L. We present a new technique for computing depth orders, and apply it to several special classes of objects. Our results include: (i) If K is a set of n triangles whose xy-projections are all `fat', then a depth order for K can be computed in time O(n log 5 n). (ii) If K is a set of n convex and simply-shaped objects whose xy-projections are all `fat' and their sizes are within a constant ratio from one another, then a depth order for K can be computed in time O(n 1=2 s (n) log 4 n), where s is the maximum number of intersections between the boundaries of the xy-projections of any pair of objects in K, and s (n) is the maximum length of (n; s) Davenport-Schinzel sequences.

. 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...

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...

We study two problems related to planar motion planning for robots with imperfect control, where, if the robot starts a linear movement in a certain commanded direction, we only know that its actual movement will be confined in a cone of angle ff centered around the specified direction. First, we consider a single goal region, namely the "region at infinity", and a set of polygonal obstacles, modeled as a set S of n line segments. We are interested in the region R ff (S) from where we can reach infinity with a directional uncertainty of ff. We prove that the maximum complexity of R ff (S) is O(n=ff 5 ). Second, we consider a collection of k polygonal goal regions of total complexity m, but without any obstacles. Here we prove an O(k 3 m) bound on the complexity of the region from where we can reach a goal region with a directional uncertainty of ff. For both situations we also prove lower bounds on the maximum complexity, and we give efficient algorithms for computing the regions. ...

Let B be a set of n arbitrary (possibly intersecting) convex obstacles in R d . It is shown that any two points which can be connected by a path avoiding the obstacles can also be connected by a path consisting of O(n (d\Gamma1)bd=2+1c ) segments. The bound cannot be improved below \Omega\Gamma n d ); thus in R 3 , the answer is between n 3 and n 4 . For open disjoint convex obstacles, a \Theta(n) bound is proved. By a well-known reduction, the general case result also upper bounds the complexity for a translational motion of an arbitrary convex robot among convex obstacles. In the planar case, asymptotically tight bounds and efficient algorithms are given. 1 Introduction The results presented in this paper are motivated by the following problem. Consider n disjoint convex obstacles of an arbitrary shape in the plane or in R 3 , and a convex robot R. Suppose that a position q of R can be reached from a position p by a translational motion of R avoiding the obstacles. I...

this paper we consider the problem of finding separators for a set of line-segments. Clearly this is sufficient to treat the case of general polygonal objects as well.