Klara Kedem

On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles

Let γ1,..., γm bem simple Jordan curves in the plane, and letK1,...,Km be their respective interior regions. It is shown that if each pair of curves γi, γj,i ≠j, intersect one another in at most two points, then the boundary ofK=∩i=1mKi contains at most max(2,6m − 12) intersection points of the curvesγ1, and this bound cannot be improved. As a corollary, we obtain a similar upper bound for the number of points of local nonconvexity in the union ofm Minkowski sums of planar convex sets. Following a basic approach suggested by Lozano Perez and Wesley, this can be applied to planning a collision-free translational motion of a convex polygonB amidst several (convex) polygonal obstaclesA1,...,Am. Assuming that the number of corners ofB is fixed, the algorithm presented here runs in timeO (n log2n), wheren is the total number of corners of theAis.

The upper envelope of voronoi surfaces and its applications

Given a setS ofsources (points or segments) in ℜ211C;d, we consider the surface in ℜ211C;d+1 that is the graph of the functiond(x)=minpεSρ(x, p) for some metricρ. This surface is closely related to the Voronoi diagram, Vor(S), ofS under the metricρ. The upper envelope of a set of theseVoronoi surfaces, each defined for a different set of sources, can be used to solve the problem of finding the minimum Hausdorff distance between two sets of points or line segments under translation. We derive bounds on the number of vertices on the upper envelope of a collection of Voronoi surfaces, and provide efficient algorithms to calculate these vertices. We then discuss applications of the methods to the problems of finding the minimum Hausdorff distance under translation, between sets of points and segments.

Computing the minimum Hausdorff distance for point sets under translation

We consider the problem of computing a translation that minimizes the Hausdorff distance between two sets of points. For points in @@@@<supscrpt>1</supscrpt> in the worst case there are ⊖(<italic>mn</italic>) translations at which the Hausdorff distance is a local minimum, where <italic>m</italic> is the number of points in one set and <italic>n</italic> is the number in the other. For points in @@@@<supscrpt>2</supscrpt> there are ⊖(<italic>mn</italic>(<italic>m</italic> + <italic>n</italic>)) such local minima. We show how to compute the minimal Hausdorff distance in time <italic>&Ogr;</italic>(<italic>mn</italic> log <italic>mn</italic>) for points in @@@@<supscrpt>1</supscrpt> and in time <italic>&Ogr;</italic>(<italic>m</italic><supscrpt>2</supscrpt><italic>n</italic><supscrpt>2</supscrpt>α(<italic>mn</italic>)) for points in @@@@<supscrpt>2</supscrpt>. The results for the one-dimensional case are applied to the problem of comparing polygons under general affine transformations, where we extend the recent results of Arkin et al on polygon resemblance under rigid body motion. The two-dimensional case is closely related to the problem of finding an approximate congruence between two points sets under translation in the plane, as considered by Alt et al.

An efficient motion-planning algorithm for a convex polygonal object in two-dimensional polygonal space

We present an efficient algorithm for planning the motion of a convex polygonal bodyB in two-dimensional space bounded by a collection of polygonal obstacles. Our algorithm extends and combines the techniques of Leven and Sharir and of Sifrony and Sharir used for the case in whichB is a line segment (a “ladder”). It also makes use of the results of Kedem and Sharir on the planning of translational motion ofB amidst polygonal obstacles, and of a recent result of Leven and Sharir on the number of free critical contacts ofB with such polygonal obstacles. The algorithm runs in timeO(knλ6(kn) logkn), wherek is the number of sides ofB, n is the number of obstacle edges, and λ,(q) is an almost linear function ofq yielding the maximal number of connected portions ofq continuous functions which compose the graph of their lower envelope, where it is assumed that each pair of these functions intersect in at mosts points.

An efficient algorithm for planning collision-free translational motion of a convex polygonal object in 2-dimensional space amidst polygonal obstacles

We state and prove a theorem about the number of points of local nonconvexity in the union of <italic>m</italic>. Minkowski sums of planar convex sets, and then apply it to planning a collision-free translational motion of a convex polygon <italic>B</italic> amidst several (convex) polygonal obstacles <italic>A</italic><subscrpt>l</subscrpt>,…, <italic>A</italic><subscrpt>m</subscrpt>, following a basic approach suggested by Lozano-Perez and Wesley. Assuming that the number of corners of <italic>B</italic> is fixed, the algorithm developed here runs in time <italic>&Ogr;</italic>(<italic>n</italic> log<supscrpt>2</supscrpt><italic>n</italic>), where <italic>n</italic> is the total number of corners of the <italic>A</italic><subscrpt><italic>l</italic></subscrpt>s.

Improvements on Geometric Pattern Matching Problems

We consider the following geometric pattern matching problem: find the minimum Hausdorff distance between two point sets under translation with L1 or L∞ as the underlying metric. Huttenlocher, Kedem, and Sharir have shown that this minimum distance can be found by constructing the upper envelope of certain Voronoi surfaces. Further, they show that if the two sets are each of cardinality n then the complexity of the upper envelope of such surfaces is Ω(n3). We examine the question of whether one can get around this cubic lower bound, and show that under the L1 and L∞ metrics, the time to compute the minimum Hausdorff distance between two point sets is On2 log2n).

A convex polygon among polygonal obstacles: placement and high-clearance motion

Abstract Given a convex polygon P and an environment consisting of polygonal obstacles, we find the placement for the largest similar copy of P that does not intersect any of the obstacles. Allowing translation, rotation, and change-of-size, our method combines a new notion of Delaunay triangulation for points and edges with the well-known functions based on Davenport–Schinzel sequences, producing an almost quadratic algorithm for the problem. Namely, if P is a convex k -gon and if Q has n corners and edges then we can find the placement of the largest similar copy of P in the environment W in time O( k 4 n λ 3 ( n )log n ), where λ 3 is one of the almost-linear functions related to Davenport–Schinzel sequences. Based on our complexity analysis of the placement problem, we develop a high-clearance motion planning technique for a convex polygonal object moving among polygonal obstacles in the plane, allowing both rotation and translation ( general motion ). Given a k -sided convex polygonal object P , a set of polygonal obstacles with n corners and edges, and given initial and final positions for P , the time needed to determine a high-clearance , obstacle-avoiding path for P is O( k 4 n λ 3 ( n )log n ).

An automatic motion planning system for a convex polygonal mobile robot in 2-dimensional polygonal space

We present an automatic system for planning the (translational and rotational) collision-free motion of a convex polygonal body <italic>B</italic> in two-dimensional space bounded by a collection of polygonal obstacles. The system consists of a (combinatorial, non-heuristic) motion planning algorithm, based on sophisticated algorithmic and combinatorial techniques in computational geometry, and is implemented on a Cartesian robot system equipped with a 2-D vision system. Our algorithm runs in the worst-case in time <italic>&Ogr;</italic>(<italic>kn</italic>λ<subscrpt>6</subscrpt>(<italic>kn</italic>) log <italic>kn</italic>), where <italic>k</italic> is the number of sides of <italic>B</italic>, <italic>n</italic> is the total number of obstacle edges, and λ<subscrpt>6</subscrpt>(<italic>r</italic>) is the (nearly-linear) maximum length of an (<italic>r</italic>, 6) Davenport Schinzel sequence. Our implemented system provides an “intelligent” robot that, using its attached vision system, can acquire a geometric description of the robot and its polygonal environment, and then, given a high-level motion command from the user, can plan a collision-free path (if one exists), and then go ahead and execute that motion.

Arrangements of segments that share endpoints: single face results

We provide new combinatorial bounds on the complexity of a face in an arrangement of segments in the plane. In particular, we show that the complexity of a single face in an arrangement ofn line segments determined byh endpoints isO(h logh). While the previous upper bound,O(nα(n)), is tight for segments with distinct endpoints, it is far from being optimal whenn=Ω(h2). Our results show that, in a sense, the fundamental combinatorial complexity of a face arises not as a result of the number ofsegments, but rather as a result of the number ofendpoints.