Kinetic collision detection between two simple polygons

AbstractWe design a kinetic data structure for detecting collisions between two simple polygons in motion. In order to do so, we create a planar subdivision of the free space between the two polygons, called the external relative geodesic triangulation, which certifies their disjointness. We show how this subdivision can be maintained as a kinetic data structure when the polygons are moving, and analyze its performance in the kinetic setting

Proximity Problems on Moving Points

A kinetic data structure for the maintenance of a multidimensional range search tree is introduced. This structure is used as a building block to obtain kinetic data structures for two classical geometric proximity problems in arbitrary dimensions: the first structure maintains the closest pair of a set of continuously moving points, and is provably e#cient. The second structure maintains a spanning tree of the moving points whose cost remains within some prescribed factor of the minimum spanning tree. The method for maintaining the closest pair of points can be extended to the maintenance of closest pair of other distance functions which allows us to maintain the closest pair of a set of moving objects with similar sizes and of a set of points on a smooth manifold

Data structures for mobile data

A kinetic data structure (KDS) maintains an attribute of interest in a system of geometric objects undergoing continuous motion. In this paper we develop a conceptual framework for kinetic data structures, propose a number of criteria for the quality of such structures, and describe a number of fundamental techniques for their design. We illustrate these general concepts by presenting kinetic data structures for maintaining the convex hull and the closest pair of moving points in the plane; these structures behavewell according to the proposed quality criteria for KDSs

Probabilistic Analysis for Combinatorial Functions of Moving Points

We initiate a probabilistic study of configuration functions of moving points. In our probabilistic model, a particle is given an initial position and a velocity drawn independently at random from the same distribution D. We show that if n particles are drawn independently at random from the uniform distribution on the square, their convex hull undergoes \Theta(log 2 n) combinatorial changes in expectation, their Voronoi diagram undergoes \Theta(n 3=2 ) combinatorial changes, and their closest pair undergoes \Theta(n) combinatorial changes. A probabilistic analysis of kinetic data structures is initiated. 1 Introduction Given a set of n points, what is the description complexity of their convex hull? In the world of analysis of algorithms, this question is understood with an implicit "in the worst case", and the answer is n bd=2c where d is the dimension of the underlying space. This is not entirely satisfactory, as this description complexity can vary tremendously depending on ..

Reporting red-blue intersections between two sets of connected line segments

Abstract. We present a new line sweep algorithm, HeapSweep, for reporting bichromatic (`purple') intersections between a red and a blue family of line segments. If the union of the segments in each family is connected as a point set, HeapSweep reports all k purple intersections in time O((n + k) (n)log 3 n), where n is the total number of input segments and (n) is the familiar inverse Ackermann function. To achieve these bounds, the algorithm keeps only partial information about the vertical ordering between segments of the same color, using a new data structure called a kinetic queue. In order to analyze the running time of HeapSweep, wealsoshow that a simple polygon containing a set of n line segments can be partitioned into monotone regions by lines cutting these segments (n log n) times.

Sweeping Lines and Line Segments with a Heap

Given n line segments in the plane, the Bentley-Ottmann sweep maintains the exact ordering of the intersections of the segments with a vertical line, as this line sweeps the plane from left to right. To accomplish this, every intersection between two segments must be processed, and the running time of the sweep can be \Omega\Gamma n 2 ). In this paper, it is shown how a heap on the intersections can be maintained during the sweep. This new type of sweep processes O(n log 2 n) intersections when sweeping over lines and O(n p n log n) intersections when sweeping over line segments. A lower bound of \Omega\Gamma n log n) is also established. 1 Introduction One of the common introductory problems in geometric algorithms is that of finding all pairwise intersections in a family of line segments in the plane. The first nontrivial solution to this problem was given by Bentley and Ottmann [BO79], who introduced in 1979 the now familiar line sweep paradigm. A vertical line is moved (`swe..