Computational Geometry
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Convex Hulls
CONVEX HULLS PETR FELKEL FEL CTU PRAGUE [email protected] https://cw.felk.cvut.cz/doku.php/courses/a4m39vg/start Based on [Berg] and [Mount] Version from 16.11.2017 Talk overview Motivation and Definitions Graham’s scan – incremental algorithm Divide & Conquer Quick hull Jarvis’s March – selection by gift wrapping Chan’s algorithm – optimal algorithm www.cguu.com Felkel: Computational geometry (2) Convex hull (CH) – why to deal with it? Shape approximation of a point set or complex shapes (other common approximations include: minimal area enclosing rectangle, circle, and ellipse,…) – e.g., for collision detection Initial stage of many algorithms to filter out irrelevant points, e.g.: – diameter of a point set – minimum enclosing convex shapes (such as rectangle, circle, and ellipse) depend only on points on CH Felkel: Computational geometry (3) Convexity Line test !!! A set S is convex convex not convex – if for any points p,q S the lines segment pq S, or – if any convex combination of p and q is in S Convex combination of points p, q is any point that can be expressed as q p =1 (1 – ) p + q, where 0 1 =0 Convex hull CH(S) of set S – is (similar definitions) – the smallest set that contains S (convex) – or: intersection of all convex sets that contain S – Or in 2D for points: the smallest convex polygon containing all given points Felkel: Computational geometry (5) Definitions from topology in metric spaces Metric space – each two of points have defined a distance r r-neighborhood of a point p and radius r> 0 p = set of -
12 Binary Space Partitions the Painter's Algorithm
Computational Geometry Third Edition Mark de Berg · Otfried Cheong Marc van Kreveld · Mark Overmars Computational Geometry Algorithms and Applications Third Edition 123 Prof. Dr. Mark de Berg Dr. Marc van Kreveld Department of Mathematics Department of Information and Computer Science and Computing Sciences TU Eindhoven Utrecht University P.O. Box 513 P.O. Box 80.089 5600 MB Eindhoven 3508 TB Utrecht The Netherlands The Netherlands [email protected] [email protected] Dr. Otfried Cheong, ne´ Schwarzkopf Prof. Dr. Mark Overmars Department of Computer Science Department of Information KAIST and Computing Sciences Gwahangno 335, Yuseong-gu Utrecht University Daejeon 305-701 P.O. Box 80.089 Korea 3508 TB Utrecht [email protected] The Netherlands [email protected] ISBN 978-3-540-77973-5 e-ISBN 978-3-540-77974-2 DOI 10.1007/978-3-540-77974-2 ACM Computing Classification (1998): F.2.2, I.3.5 Library of Congress Control Number: 2008921564 © 2008, 2000, 1997 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. -
Acknowledgements
Acknowledgements So many people have made a significant contribution to this thesis in various ways. Al- though all of them provided me with valuable information, a few played a more direct role in the preparation of this thesis. First of all, I extend my gratitude to my promotor, Mark Overmars, and my co-promotor, Otfried Cheong for their perfect supervision. During the writing of my Ph.D. thesis, Mark Overmars provided many corrections and suggestions for improvement of the manuscript, for which I’m grateful. I am deeply indebted to Otfried Cheong: When I was a M.Sc. student at POSTECH in Korea, he motivated me to do a Ph.D. in Computational Geometry and provided me an offer of being his first Ph.D. student. With unfailing courtesy and monumental patience, he have supervised me in a perfect way, in Hong Kong and the Netherlands. I would also like to thank Siu-Wing Cheng and Mordecai Golin. Siu-Wing Cheng had advised me in various ways, and we had worked together on several problems. Mordecai Golin had showed me interesting geometry problems that I challenged to solve. I should also thank Rene´ van Oostrum. When we were in Hong Kong, I learned from him how to develope films and print photographs. He also helped me a lot when I settled down to the life in the Netherlands, and translated “Summary” into Dutch for this thesis. I must acknowledge my debt to the co-authors of the papers of which this thesis is com- posed: apart from Otfried Cheong, Siu-Wing Cheng and Rene´ van Oostrum, these are Mark de Berg, Prosenjit Bose, Dan Halperin, Jirˇ´ı Matousek,ˇ and Jack Snoeyink. -
3SUM with Preprocessing
Data Structures Meet Cryptography: 3SUM with Preprocessing Alexander Golovnev Siyao Guo Thibaut Horel Harvard NYU Shanghai MIT [email protected] [email protected] [email protected] Sunoo Park Vinod Vaikuntanathan MIT & Harvard MIT [email protected] [email protected] Abstract This paper shows several connections between data structure problems and cryptography against preprocessing attacks. Our results span data structure upper bounds, cryptographic applications, and data structure lower bounds, as summarized next. First, we apply Fiat{Naor inversion, a technique with cryptographic origins, to obtain a data-structure upper bound. In particular, our technique yields a suite of algorithms with space S and (online) time T for a preprocessing version of the N-input 3SUM problem where S3 · T = Oe(N 6). This disproves a strong conjecture (Goldstein et al., WADS 2017) that there is no data structure that solves this problem for S = N 2−δ and T = N 1−δ for any constant δ > 0. Secondly, we show equivalence between lower bounds for a broad class of (static) data struc- ture problems and one-way functions in the random oracle model that resist a very strong form of preprocessing attack. Concretely, given a random function F :[N] ! [N] (accessed as an oracle) we show how to compile it into a function GF :[N 2] ! [N 2] which resists S-bit prepro- cessing attacks that run in query time T where ST = O(N 2−") (assuming a corresponding data structure lower bound on 3SUM). In contrast, a classical result of Hellman tells us that F itself can be more easily inverted, say with N 2=3-bit preprocessing in N 2=3 time. -
Jean-Paul Laumond
San Francisco CA 1:51:39 m4v Jean-Paul Laumond An interview conducted by Selma Šabanović with Matthew R. Francisco September 27 2011 Q: So if we can start with your name, and where you were born and when? Jean-Paul Laumond: Okay I'm Jean-Paul Laumond. I born in '53 in countryside North from Toulouse in France, in a small village in France. Q: And where did you go to school? Jean-Paul Laumond: I have been to school in Brive which is the main city of the area in that countryside, and after that I entered the university in Toulouse, first by following special training in French which is what we call the “classe préparatoire” for the engineering school. But in fact after two years I moved to university to in the mathematics department and I got a diploma to be a teacher in mathematics. Q: What did you study while you were in university and how did you decide on that particular direction? Jean-Paul Laumond: <clears throat> At the very beginning, I was not very convinced by the choice I made to enter into engineering, because engineering is to go towards the industry world while I was more interested by the academic position. And at that time I didn't know that engineering could be a possibility to enter the university or to make then. Then for me it was just pure abstraction or like literature, and then this is why and of course I was very enthusiastic with mathematics, and this is why I chosen to enter in the university in the department of mathematics. -
Creative and Coordinated Computation
Creative and Coordinated Computation Inaugural address delivered by Marc van Kreveld Professor in Computational Geometry and its Application Department of Information and Computing Sciences Faculty of Science Utrecht University on Friday, January 17, 2014 Utrecht, The Netherlands c Marc van Kreveld, 2014 Rector magnificus, colleagues, family, friends, ... I am honored to stand before you today, and have the opportunity to tell you all about my work, my research, and some other things that I will get to later. It is Friday afternoon, after 4 o'clock, and I realize that most people would call this the beginning of the weekend. However, I am going to be talking about scientific work for a while, so bear with me. I promise that I will not show any formulas, but I will show many illustrations. I truly hope that everyone in the audience will hear something worthwhile in this lecture. My research is related to geometry, a branch of mathematics that deals with objects, like points, lines, triangles and circles, and addresses rela- tions between these objects, like distances, intersections, and containment, or properties like lengths, areas, and angles. My research is also about al- gorithms, the use of automated methods to execute tasks, and the formal study of these methods. The third aspect of my research is applications, because points, triangles and circles can have a concrete meaning in appli- cations only, and the application dictates which tasks we want to perform on the geometry. The title of this speech, \Creative and Coordinated Computation", may seem contradictory: a creative process is often not coordinated, and more- over, coordination may obstruct creativity. -
Geometric Problems in Cartographic Networks
Geometric Problems in Cartographic Networks Geometrische Problemen in Kartografische Netwerken (met een samenvatting in het Nederlands) PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. Dr. W.H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op maandag 29 maart 2004 des middags te 16:15 uur door Sergio Cabello Justo geboren op 1 december 1977, te Lleida Promotor: Prof. Dr. Mark H. Overmars Copromotor: Dr. Marc J. van Kreveld Faculteit Wiskunde en Informatica Universiteit Utrecht Advanced School for Computing and Imaging This work was carried out in the ASCI graduate school. ASCI dissertation series number 99. This work has been supported mainly by the Cornelis Lely Stichting, and par- tially by a Marie Curie Fellowship of the European Community programme \Combinatorics, Geometry, and Computation" under contract number HPMT- CT-2001-00282, and a NWO travel grant. Dankwoord / Agradecimientos It is my wish to start giving my deepest thanks to my supervisor Marc van Kreveld and my promotor Mark Overmars. I am also very grateful to Mark de Berg, who also participated in my supervision in the beginning. Without trying it, you made me think that I was in the best place to do my thesis. I would also like to thank Pankaj Agarwal, Mark de Berg, Jan van Leeuwen, Gun¨ ter Rote, and Dorothea Wagner for taking part in the reading committee and giving comments. I would also like to thank the co-authors who directly par- ticipate in this thesis: Mark de Berg, Erik Demaine, Steven van Dijk, Yuanxin (Leo) Liu, Andrea Mantler, Gun¨ ter Rote, Jack Snoeyink, Tycho Strijk, and, of course, Marc van Kreveld. -
Computational Geometry
Computational Geometry Third Edition Mark de Berg · Otfried Cheong Marc van Kreveld · Mark Overmars Computational Geometry Algorithms and Applications Third Edition 123 Prof. Dr. Mark de Berg Dr. Marc van Kreveld Department of Mathematics Department of Information and Computer Science and Computing Sciences TU Eindhoven Utrecht University P.O. Box 513 P.O. Box 80.089 5600 MB Eindhoven 3508 TB Utrecht The Netherlands The Netherlands [email protected] [email protected] Dr. Otfried Cheong, ne´ Schwarzkopf Prof. Dr. Mark Overmars Department of Computer Science Department of Information KAIST and Computing Sciences Gwahangno 335, Yuseong-gu Utrecht University Daejeon 305-701 P.O. Box 80.089 Korea 3508 TB Utrecht [email protected] The Netherlands [email protected] ISBN 978-3-540-77973-5 e-ISBN 978-3-540-77974-2 DOI 10.1007/978-3-540-77974-2 ACM Computing Classification (1998): F.2.2, I.3.5 Library of Congress Control Number: 2008921564 © 2008, 2000, 1997 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. -
Algorithms for Robotic Motion and Manipulation Taylor &Francis Taylor & Francis Group Algorithms for Robotic Motion and Manipulation
Algorithms for Robotic Motion and Manipulation Taylor &Francis Taylor & Francis Group http://taylorandfrancis.com Algorithms for Robotic Motion and Manipulation 1996 Workshop on the Algorithmic Foundations of Robotics edited, by Jean-Paul Laumond LAAS-CNRS Toulouse, France Mark Overmars Utrecht University Utrecht, The Netherlands Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business First published 1997 by AK Peters Ltd. Published 2018 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1997 by Taylor & Francis Group, LLC CRC Press is an imprint ofTaylor & Francis Group, an Informa business No claim to original U.S. Government works ISBN-13: 978-1-56881-067-6 (hbk) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. -
Center Point of Simple Area Feature Based on Triangulation Skeleton Graph
Center Point of Simple Area Feature Based on Triangulation Skeleton Graph Wei Lu Wuhan University, Wuhan, China [email protected] https://orcid.org/0000-0002-9703-2871 Tinghua Ai Wuhan University, Wuhan, China [email protected] https://orcid.org/0000-0002-6581-9872 Abstract In the area of cartography and geographic information science, the center points of area features are related to many fields. The centroid is a conventional choice of center point of area feature. However, it is not suitable for features with a complex shape for the center point may be outside the area or not fit the visual center so well. This paper proposes a novel method to calculate the center point of area feature based on triangulation skeleton graph. This paper defines two kinds of centrality of vertices in skeleton graph according to the centrality theory in graph and network analysis. Through the measurement of vertices centrality, the center points of polygon area features are defined as the vertices with maximum centrality. 2012 ACM Subject Classification Information systems → Geographic information systems Keywords and phrases Shape Center, Triangulation Skeleton Graph, Graph Centrality Digital Object Identifier 10.4230/LIPIcs.GIScience.2018.41 Category Short Paper Funding This research was supported by the National Key Research and Development Program of China (Grant No. 2017YFB0503500), and the National Natural Science Foundation of China (Grant No. 41531180). 1 Introduction In geographic information science (GIS), skeleton and center point are two important abstract descriptors of area feature which are extensively used in spatial data compression, cartographic generalization, map annotation configuration, multiscale map matching, spatial relation calculation, etc. -
Geometric Algorithms for Geographic Information Systems
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Utrecht University Repository Geometric Algorithms for Geographic Information Systems Geometrische Algoritmen voor Geografische Informatiesystemen (met een samenvatting in het Nederlands) PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. Dr. H.O. Voorma ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op maandag 31 mei 1999 des middags te 12:45 uur door Rene´ Willibrordus van Oostrum geboren op 7 november 1965, te Utrecht promotor: Prof. Dr. M.H. Overmars Faculteit Wiskunde en Informatica co-promotor: Dr. M.J. van Kreveld Faculteit Wiskunde en Informatica ISBN 90-393-2049-7 This research was partially supported by the ESPRIT IV LTR Project No. 21957 (CGAL). Contents 1 Introduction 1 1.1 Geography: preliminaries .......................... 2 1.1.1 Geography and related disciplines ................. 2 1.1.2 Geographic data .......................... 3 1.1.3 Geographic maps .......................... 4 1.2 GIS: preliminaries .............................. 4 1.2.1 GIS functionality .......................... 4 1.2.2 GIS data models and structures .................. 8 1.3 Computational geometry: preliminaries ................... 12 1.3.1 The DCEL structure ......................... 13 1.3.2 The sweeping paradigm ...................... 16 1.3.3 Search structures .......................... 19 1.3.4 Voronoi diagrams .......................... 22 1.4 Selected computational geometry problems from GIS ........... 24 1.4.1 Choropleth map traversal ...................... 24 1.4.2 Isosurface generation ........................ 25 1.4.3 Selection for cartographic generalization .............. 26 1.4.4 Facility location in terrains ..................... 27 2 Subdivision Traversal Without Extra Storage 29 2.1 Introduction ................................ -
Computational Geometry Lectures
Computational Geometry Lectures Olivier Devillers & Monique Teillaud Bibliography Books [30, 19, 18, 45, 66, 70] 1 Convex hull: definitions, classical algorithms. • Definition, extremal point. [66] • Jarvis algorithm. [54] • Orientation predicate. [55] • Buggy degenerate example. [36] • Real RAM model and general position hypothesis. [66] • Graham algorithm. [50] • Lower bound. [66] • Other results. [67, 68, 58, 56, 8, 65, 7] • Higher dimensions. [82, 25] 2 Delaunay triangulation: definitions, motivations, proper- ties, classical algorithms. • Space of spheres [31, 32] • Delaunay predicates. [42] • Flipping trianagulation [52] • Incremental algorithm [57] • Sweep-line [48] • Divide&conquer [74] • Deletion [35, 20, 37] 1 3 Probabilistic analyses: randomized algorithms, evenly dis- tributed points. • Randomization [29, 73, 14] – Delaunay tree [16, 17] and variant [51] – Jump and walk [63, 62] – Delaunay hierarchy [34] – Biased random insertion order [6] – Accelerated algorithms [28, 72, 33, 26] • Poisson point processes [71, 64] – Straight walk [44, 27] – Visibility walk [39] – Walks on vertices [27, 41, 59] – Smoothed analysis [75, 38] 4 Robustness issues: numerical issues, degenerate cases. • IEEE754 [49] • Orientation with double [55] • Solution 1: no geometric theorems [76] • Exact geometric computation paradigm [81] • Perturbations – controlled [61] – symbolic: SoS [46], world [2], 4th dim [43], “qualitative” [40] 5 Triangulations in the CGAL library. • https://doc.cgal.org/latest/Manual/packages.html#PartTriangulationsAndDelaunayTriangulations