Let P be a set of polygonal pseudodiscs in the plane with n edges in total translating with fixed velocities in fixed directions. We prove that the maximum number of combinatorial changes in the union of P is \Theta(n 2 ff(n)). In general, if the pseudodiscs move along curved trajectories, then the maximum number of changes in the union is \Theta(n s+2 (n)), where s is the maximum number of times any triple of polygon edges meet in a common point. We apply this result in two different settings. First, we prove that the complexity of the free space of a constant-complexity polygon translating amidst convex polyhedral obstacles with n edges in total is O(n 2 ff(n)). Second, we show that the complexity of the space of lines missing a set of n convex homothetic polytopes of constant complexity in 3-space is O(n 2 4 (n)). Both bounds are almost tight in the worst case. 1 Introduction Let P be a set of polygons in the plane with n edges in total. Each polygon translates with a fixed ...

We discuss a technical problem arising in the motion planning algorithm of Kedem and Sharir [KS], and propose a way to overcome it without increasing the asymptotic complexity of the algorithm. 1 Introduction The paper "An efficient motion-planning algorithm for a convex polygonal object in two-dimensional polygonal space", by Kedem and Sharir [KS], studies the problem of planning a collision-free motion (including translation and rotation) for a convex polygonal body B, with k corners, amidst polygonal obstacles having n corners altogether. More specifically, the problem is stated as follows: given initial and final placements of B, determine whether there is an obstacle-avoiding motion from the initial placement to the final placement and, if so, plan such a motion. In what follows we assume some familiarity of the reader with the algorithm of [KS]. Nevertheless we will present a brief description of the technique, providing enough details to allow us to state the technical diff...

We study two problems involving collision-free placements of a convex m-gon P in a planar polygonal environment: (i) We first show that the largest similar copy of P inside another convex polygon Q with n edges can be computed in O(mn 2 log n) time. We also show that the combinatorial complexity of the space of all similar copies of P inside Q is O(mn 2 ), and that it can also be computed in O(mn 2 log n) time. (ii) We then consider the case where Q is an arbitrary polygonal environment with n edges. We give the first (and relatively simple) 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- Pankaj Agarwal has been supported by NSF Grant CCR93 --01259, an NYI award, and by matching funds from Xerox Corp. Nina Amenta has been supported by the Geometry Center, which is officially the Center for Computation and Visualization of Geometric Structures, supported by NSF/DMS8920161. Boris Aro...

We study the space of free translations of a box amidst polyhedral obstacles with n vertices. We show that the combinatorial complexity of this space is O(n 2 ff(n)) where ff(n) is the inverse Ackermann function. Our bound is within an ff(n) factor off the lower bound, and it constitutes an improvement of almost an order of magnitude over the best previously known (and naive) bound for this problem, O(n 3 ). For the case of a convex polygon of fixed (constant) size translating in the same setting (namely, a two-dimensional polygon translating in three-dimensional space), we show a tight bound \Theta(n 2 ff(n)) on the complexity of the free space. A preliminary version of this paper appeared in Proc. 9th ACM Symposium on Computational Geometry, San Diego, 1993. Work on this paper by the first author has been supported by a Rothschild Postdoctoral Fellowship, by a grant from the Stanford Integrated Manufacturing Association (SIMA), by NSF/ARPA Grant IRI-9306544, and by NSF Grant...

Abstract. Let O = {O1,..., Om} be a set of m convex polygons in R 2 with a total of n vertices, and let B be another convex k-gon. A placement of B, any congruent copy of B (without reflection), is called free if B does not intersect the interior of any polygon in O at this placement. A placement z of B is called critical if B forms three “distinct ” contacts with O at z. Let ϕ(B,O) be the number of free critical placements. A set of placements of B is called a stabbing set of O if each polygon in O intersects at least one placement of B in this set. We develop efficient Monte Carlo algorithms that compute a stabbing set of size h = O(h ∗ log m), with high probability, where h ∗ is the size of the optimal stabbing set of O. We also improve bounds on ϕ(B, O) for the following three cases, namely, (i) B is a line segment and the obstacles in O are pairwise-disjoint, (ii) B is a line segment and the obstacles in O may intersect (iii) B is a convex k-gon and the obstacles in O are disjoint, and use these improved bounds to analyze the running time of our stabbing-set algorithm. 1

Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in robotics, molecular modeling, solid modeling, and shape matching, and the techniques used for their solutions are interesting in their own right.

A dihedral (trihedral) wedge is the intersection of two (resp. three) half-spaces in R³. It is called -fat if the angle (resp., solid angle) determined by these half-spaces is at least > 0. If, in addition, the sum of the three face angles of a trihedral wedge is at least > 4 =3, then it is called (; )-substantially fat. We prove that, for any fixed > 4 =3; > 0, the combinatorial complexity of the union of n (a) -fat dihedral wedges, (b) (; )-substantially fat trihedral wedges is at most O(n 2+" ), for any " > 0, where the constants of proportionality depend on ", (and ). We obtain as a corollary that the same upper bound holds for the combinatorial complexity of the union of n (nearly) congruent cubes in R³. These bounds are not far from being optimal.