Computational Geometry

ACM Transactions on Graphics, 1984

Proceedings of the twentieth annual symposium on Computational geometry - SCG '04, 2004

Page 1. Towards In-Place Geometric Algorithms and Data Structures Hervé Brönnimann∗ Computer & Information Sci. Dept. Polytechnic University [email protected] Timothy M. Chan† School of Computer Science University of Waterloo [email protected] Eric Y. Chen ...

1998

2009

From March 8 to March 13, 2009, the Dagstuhl Seminar 09111 Computational Geometry was held in Schloss Dagstuhl Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The rst section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available. 09111 Executive Summary Computational Geometry 1 Computational Geometry Evolution The eld of computational geometry is concerned with the design, analysis, and implementation of algorithms for geometric problems, which arise in a wide range of areas, including computer graphics, CAD, robotics computer vision, image processing, spatial databases, GIS, molecular biology, and sensor networks. Since the mid 1980s, computational geometry has arisen as a...

Handbooks in Operations Research and Management Science, 1995

2018

Ply number is a recently developed graph drawing metric inspired by studying road networks. Informally, for each vertex v, which is associated with a point in the plane, a disk is drawn centered on v with a radius that is α times the length of the longest edge incident to v, for some constant α ∈ (0, 0.5]. The ply number is the maximum number of disks that overlap at a single point. We show that any tree with maximum degree ∆ has a 1-ply drawing when α = O(1/∆). We also show that trees can be drawn with logarithmic ply number (for α = 0.5), with an area that is polynomial for boundeddegree trees. Lastly, we show that this logarithmic upper bound does not apply to 2-trees, by giving a lower bound of Ω( √ n/ log n) ply.

mut Alt (FU Berlin), Bernard Chazelle (Princeton University) and Emo Welzl (FU Berlin). The 31 participants came from 8 countries, 12 of them came from North America and Israel. 29 lectures were given at the seminar, covering quite a number of topics in computational geometry. Unlike last year, there was no special concentration on any subject. In fact, there were talks on graph algorithms, parallel algorithms, motion planning, application-oriented problems, numerical robustness, similarity and congruence, randomized algorithms, dynamic algorithms, and a talk on implementations. As last year, an open problem session was held on Monday evening, chaired by Micha Sharir. It was stated that most of the problem discussed in last year's session had been solved (or at least some progress had been made). Let us hope that this yearr s session (reported here by Micha Sharir) will prove as fruitful. Berichterstatter: Otfried Schwarzkopf Participants: Helmut Alt, Freie

2016

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(nα(n) log n) time (where α is the inverse Ackermann function), improving Bokal et al.'s O(n log 2 n) and O(n log n log log n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems. 1998 ACM Subject Classification F.2.2 [Nonnumerical Algorithms and Problems] Geometrical problems and computations

Computational Geometry, 1998

In this paper we describe and discuss a kernel for higher-dimensional computational geometry and we present its application in the calculation of convex hulls and Delaunay triangulations. The kernel is available in form of a software library module programmed in C++ extending LEDA. We introduce the basic data types like points, vectors, directions, hyperplanes, segments, rays, lines, spheres, affine transformations, and operations connecting these types. The description consists of a motivation for the basic class layout as well as topics like layered software design, runtime correctness via checking routines and documentation issues. Finally we shortly describe the usage of the kernel in the application domain.

ACM SIGGRAPH Computer Graphics, 1987

A new class of art gallery-like problems inspired by wireless localization is discussed. An interesting new variant of classic art gallery theorems has emerged from wireless localization, as described in a paper by Eppstein, Goodrich, and Sitchinava [EGS06]. The geometric problem may be phrased as follows. A "virtual" polygon P of n vertices is given. A number g of point stations are placed in the plane, each of which broadcasts a unique index (or "key") within a fixed angular range. P is virtual in the sense that it does not block broadcasts. The goal is to place and orient stations so that each point in the plane can determine if it is in or out of P from a monotone Boolean formula (and (•) or (+) operations only) composed from the broadcasts. For example, the pentagon in Fig. 1a is determined by the formula A•B•C, and the 12-vertex polygon in Fig. 1b is determined by