Prove That the Dihedral Angles of a Regular Polyhedron Are Identical
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Sylvester-Gallai-Like Theorems for Polygons in the Plane
RC25263 (W1201-047) January 24, 2012 Mathematics IBM Research Report Sylvester-Gallai-like Theorems for Polygons in the Plane Jonathan Lenchner IBM Research Division Thomas J. Watson Research Center P.O. Box 704 Yorktown Heights, NY 10598 Research Division Almaden - Austin - Beijing - Cambridge - Haifa - India - T. J. Watson - Tokyo - Zurich LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g. , payment of royalties). Copies may be requested from IBM T. J. Watson Research Center , P. O. Box 218, Yorktown Heights, NY 10598 USA (email: [email protected]). Some reports are available on the internet at http://domino.watson.ibm.com/library/CyberDig.nsf/home . Sylvester-Gallai-like Theorems for Polygons in the Plane Jonathan Lenchner¤ Abstract Given an arrangement of lines in the plane, an ordinary point is a point of intersection of precisely two of the lines. Motivated by a desire to understand the fine structure of ordinary points in line arrangements, we consider the following problem: given a polygon P in the plane and a family of lines passing into the interior of P, how many ordinary intersection points must there be on or inside of P? We answer this question for a variety of different types of polygons. -
Stony Brook University
SSStttooonnnyyy BBBrrrooooookkk UUUnnniiivvveeerrrsssiiitttyyy The official electronic file of this thesis or dissertation is maintained by the University Libraries on behalf of The Graduate School at Stony Brook University. ©©© AAAllllll RRRiiiggghhhtttsss RRReeessseeerrrvvveeeddd bbbyyy AAAuuuttthhhooorrr... Combinatorics and Complexity in Geometric Visibility Problems A Dissertation Presented by Justin G. Iwerks to The Graduate School in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics and Statistics (Operations Research) Stony Brook University August 2012 Stony Brook University The Graduate School Justin G. Iwerks We, the dissertation committee for the above candidate for the Doctor of Philosophy degree, hereby recommend acceptance of this dissertation. Joseph S. B. Mitchell - Dissertation Advisor Professor, Department of Applied Mathematics and Statistics Esther M. Arkin - Chairperson of Defense Professor, Department of Applied Mathematics and Statistics Steven Skiena Distinguished Teaching Professor, Department of Computer Science Jie Gao - Outside Member Associate Professor, Department of Computer Science Charles Taber Interim Dean of the Graduate School ii Abstract of the Dissertation Combinatorics and Complexity in Geometric Visibility Problems by Justin G. Iwerks Doctor of Philosophy in Applied Mathematics and Statistics (Operations Research) Stony Brook University 2012 Geometric visibility is fundamental to computational geometry and its ap- plications in areas such as robotics, sensor networks, CAD, and motion plan- ning. We explore combinatorial and computational complexity problems aris- ing in a collection of settings that depend on various notions of visibility. We first consider a generalized version of the classical art gallery problem in which the input specifies the number of reflex vertices r and convex vertices c of the simple polygon (n = r + c). -
Preprocessing Imprecise Points and Splitting Triangulations
Preprocessing Imprecise Points and Splitting Triangulations Marc van Kreveld Maarten L¨offler Joseph S.B. Mitchell Technical Report UU-CS-2009-007 March 2009 Department of Information and Computing Sciences Utrecht University, Utrecht, The Netherlands www.cs.uu.nl ISSN: 0924-3275 Department of Information and Computing Sciences Utrecht University P.O. Box 80.089 3508 TB Utrecht The Netherlands Preprocessing Imprecise Points and Splitting Triangulations∗ Marc van Kreveld Maarten L¨offler Joseph S.B. Mitchell [email protected] [email protected] [email protected] Abstract Traditional algorithms in computational geometry assume that the input points are given precisely. In practice, data is usually imprecise, but information about the imprecision is often available. In this context, we investigate what the value of this information is. We show here how to preprocess a set of disjoint regions in the plane of total complexity n in O(n log n) time so that if one point per set is specified with precise coordinates, a triangulation of the points can be computed in linear time. In our solution, we solve another problem which we believe to be of independent interest. Given a triangulation with red and blue vertices, we show how to compute a triangulation of only the blue vertices in linear time. 1 Introduction Computational geometry deals with computing structure in 2-dimensional space (or higher). The most popular input is a set of points, on which something useful is then computed, for example, a triangulation. Algorithms for these tasks have been developed many years ago, and are provably fast and correct. -
Computational Geometry Lecture Notes1 HS 2012
Computational Geometry Lecture Notes1 HS 2012 Bernd Gärtner <[email protected]> Michael Hoffmann <[email protected]> Tuesday 15th January, 2013 1Some parts are based on material provided by Gabriel Nivasch and Emo Welzl. We also thank Tobias Christ, Anna Gundert, and May Szedlák for pointing out errors in preceding versions. Contents 1 Fundamentals 7 1.1 ModelsofComputation ............................ 7 1.2 BasicGeometricObjects. 8 2 Polygons 11 2.1 ClassesofPolygons............................... 11 2.2 PolygonTriangulation . 15 2.3 TheArtGalleryProblem ........................... 19 3 Convex Hull 23 3.1 Convexity.................................... 24 3.2 PlanarConvexHull .............................. 27 3.3 Trivialalgorithms ............................... 29 3.4 Jarvis’Wrap .................................. 29 3.5 GrahamScan(SuccessiveLocalRepair) . .... 31 3.6 LowerBound .................................. 33 3.7 Chan’sAlgorithm................................ 33 4 Line Sweep 37 4.1 IntervalIntersections . ... 38 4.2 SegmentIntersections . 38 4.3 Improvements.................................. 42 4.4 Algebraic degree of geometric primitives . ....... 42 4.5 Red-BlueIntersections . .. 45 5 Plane Graphs and the DCEL 51 5.1 TheEulerFormula............................... 52 5.2 TheDoubly-ConnectedEdgeList. .. 53 5.2.1 ManipulatingaDCEL . 54 5.2.2 GraphswithUnboundedEdges . 57 5.2.3 Remarks................................. 58 3 Contents CG 2012 6 Delaunay Triangulations 61 6.1 TheEmptyCircleProperty . 64 6.2 TheLawsonFlipalgorithm -
5 PSEUDOLINE ARRANGEMENTS Stefan Felsner and Jacob E
5 PSEUDOLINE ARRANGEMENTS Stefan Felsner and Jacob E. Goodman INTRODUCTION Pseudoline arrangements generalize in a natural way arrangements of straight lines, discarding the straightness aspect, but preserving their basic topological and com- binatorial properties. Elementary and intuitive in nature, at the same time, by the Folkman-Lawrence topological representation theorem (see Chapter 6), they provide a concrete geometric model for oriented matroids of rank 3. After their explicit description by Levi in the 1920’s, and the subsequent devel- opment of the theory by Ringel in the 1950’s, the major impetus was given in the 1970’s by Gr¨unbaum’s monograph Arrangements and Spreads, in which a number of results were collected and a great many problems and conjectures posed about arrangements of both lines and pseudolines. The connection with oriented ma- troids discovered several years later led to further work. The theory is by now very well developed, with many combinatorial and topological results and connections to other areas as for example algebraic combinatorics, as well as a large number of applications in computational geometry. In comparison to arrangements of lines arrangements of pseudolines have the advantage that they are more general and allow for a purely combinatorial treatment. Section 5.1 is devoted to the basic properties of pseudoline arrangements, and Section 5.2 to related structures, such as arrangements of straight lines, configura- tions (and generalized configurations) of points, and allowable sequences of permu- tations. (We do not discuss the connection with oriented matroids, however; that is included in Chapter 6.) In Section 5.3 we discuss the stretchability problem. -
Chapter 12 Pseudotriangulations
Chapter 12 Pseudotriangulations We have seen that arrangements and visibility graphs are useful and powerful models for motion planning. However, these structures can be rather large and thus expensive to build and to work with. Let us therefore consider a simpli ed problem: for a robot which is positioned in some environment of obstacles, which and where is the next obstacle that it would hit, if it continues to move linearly in some direction? This type of problem is known as a ray shooting query because we imagine to shoot a ray starting from the current position in a certain direction and want to know what is the rst object hit by this ray. If the robot is modeled as a point and the environment as a simple polygon, we arrive at the following problem. Problem 8 (Ray-shooting in a simple polygon.) Given a simple polygon P = (p1, . , pn), a point q 2 R2, and a ray r emanating from q, which is the rst edge of P hit by r? r q P Figure 12.1: An instance of the ray-shooting problem within a simple polygon. In the end, we would like to have a data structure to preprocess a given polygon such that a ray shooting query can be answered eciently for any query ray starting somewhere inside P. As a warmup, let us look at the case that P is a convex polygon. Here the problem is easy: Supposing we are given the vertices of P in an array-like structure, the boundary 103 Chapter 12. -
The Symplectic Geometry of Polygons in Euclidean Space
The Symplectic Geometry of Polygons in Euclidean Space Michael Kap ovich and John J Millson March Abstract We study the symplectic geometry of mo duli spaces M of p oly r gons with xed side lengths in Euclidean space We show that M r has a natural structure of a complex analytic space and is complex 2 n analytically isomorphic to the weighted quotient of S constructed by Deligne and Mostow We study the Hamiltonian ows on M ob r tained by b ending the p olygon along diagonals and show the group generated by such ows acts transitively on M We also relate these r ows to the twist ows of Goldman and JereyWeitsman Contents Intro duction Mo duli of p olygons and weighted quotients of conguration spaces of p oints on the sphere Bending ows and p olygons Actionangle co ordinates This research was partially supp orted by NSF grant DMS at University of Utah Kap ovich and NSF grant DMS the University of Maryland Millson The connection with gauge theory and the results of Gold man and JereyWeitsman Transitivity of b ending deformations Bending of quadrilaterals Deformations of ngons Intro duction Let P b e the space of all ngons with distinguished vertices in Euclidean n 3 space E An ngon P is determined by its vertices v v These vertices 1 n are joined in cyclic order by edges e e where e is the oriented line 1 n i segment from v to v Two p olygons P v v and Q w w i i+1 1 n 1 n are identied if and only if there exists an orientation preserving isometry g 3 of E which -
GEOMETRIC FOLDING ALGORITHMS I
P1: FYX/FYX P2: FYX 0521857570pre CUNY758/Demaine 0 521 81095 7 February 25, 2007 7:5 GEOMETRIC FOLDING ALGORITHMS Folding and unfolding problems have been implicit since Albrecht Dürer in the early 1500s but have only recently been studied in the mathemat- ical literature. Over the past decade, there has been a surge of interest in these problems, with applications ranging from robotics to protein folding. With an emphasis on algorithmic or computational aspects, this comprehensive treatment of the geometry of folding and unfolding presents hundreds of results and more than 60 unsolved “open prob- lems” to spur further research. The authors cover one-dimensional (1D) objects (linkages), 2D objects (paper), and 3D objects (polyhedra). Among the results in Part I is that there is a planar linkage that can trace out any algebraic curve, even “sign your name.” Part II features the “fold-and-cut” algorithm, establishing that any straight-line drawing on paper can be folded so that the com- plete drawing can be cut out with one straight scissors cut. In Part III, readers will see that the “Latin cross” unfolding of a cube can be refolded to 23 different convex polyhedra. Aimed primarily at advanced undergraduate and graduate students in mathematics or computer science, this lavishly illustrated book will fascinate a broad audience, from high school students to researchers. Erik D. Demaine is the Esther and Harold E. Edgerton Professor of Elec- trical Engineering and Computer Science at the Massachusetts Institute of Technology, where he joined the faculty in 2001. He is the recipient of several awards, including a MacArthur Fellowship, a Sloan Fellowship, the Harold E. -
Pseudotriangulations and the Expansion Polytope
99 Pseudotriangulations G¨unter Rote Freie Universit¨atBerlin, Institut f¨urInformatik 2nd Winter School on Computational Geometry Tehran, March 2–6, 2010 Day 5 Literature: Pseudo-triangulations — a survey. G.Rote, F.Santos, I.Streinu, 2008 98 Outline 1. Motivation: ray shooting 2. Pseudotriangulations: definitions and properties 3. Rigidity, Laman graphs 4. Rigidity: kinematics of linkages 5. Liftings of pseudotriangulations to 3 dimensions 97 1. Motivation: Ray Shooting in a Simple Polygon 97 1. Motivation: Ray Shooting in a Simple Polygon Walking in a triangulation: Walk to starting point. Then walk along the ray. 97 1. Motivation: Ray Shooting in a Simple Polygon Walking in a triangulation: Walk to starting point. Then walk along the ray. O(n) steps in the worst case. 96 Triangulations of a convex polygon 1 12 2 11 3 10 4 9 5 8 6 7 96 Triangulations of a convex polygon 1 1 12 2 12 2 11 3 11 3 10 4 10 4 9 5 9 5 8 6 8 6 7 7 balanced triangulation A path crosses O(log n) triangles. 95 Triangulations of a simple polygon 1 5 2 12 1 12 2 3 11 10 11 4 3 6 8 10 4 9 7 9 5 balanced triangulation: balanced geodesic8 triangulation:6 An edge crosses O(log n) An edge crosses O7 (log n) triangles. pseudotriangles. [Chazelle, Edelsbrunner, Grigni, Guibas, Hershberger, Sharir, Snoeyink 1994] 95 Triangulations of a simple polygon 1 5 2 12 1 12 2 3 11 10 11 4 3 corner 6 8 pseudotriangle 10 4 tail 9 7 9 5 balanced triangulation: balanced geodesic8 triangulation:6 An edge crosses O(log n) An edge crosses O7 (log n) triangles. -
© Cambridge University Press Cambridge
Cambridge University Press 978-0-521-85757-4 - Geometric Folding Algorithms: Linkages, Origami, Polyhedra Erik D. Demaine and Joseph O’Rourke Index More information Index 1-skeleton, 311, 339 bar, 9 3-Satisfiability, 217, 221 base, see origami, base, 2 α-cone canonical configuration, 151, 152 Bauhaus, 294 α-producible chain, 150, 151 Bellows theorem, 279, 348 δ-perturbation, 115 bending λ order function, 176, 177, 186 machine, xi, 13, 306 antisymmetry condition, 177, 186 pipe, 13, 14 consistency condition, 178, 186 sheet metal, 306 noncrossing condition, 179, 186 beta sheet, 158 time continuity, 174, 183, 187 Bezdek, Daniel, 331 transitivity condition, 178, 186 blooming, continuous, 333, 435 bond angle, 14, 131, 148, 151 Abe’s angle trisection, 286, 287 bond length, 148 accordion, 85, 193, 200, 261 active path, 244, 245, 247–249 cable, 53–55 acyclicity, 108 CAD, see cylindrical algebraic decomposition, 19 additor (Kempe), 32, 34, 35 cage, 21, 92, 93 Alexandrov, Aleksandr D., 348 canonical form, 74, 86, 87, 141, 151 Alexandrov’s theorem, 339, 348, 349, 352, 354, Cauchy’s arm lemma, 72, 133, 143, 145, 342, 343, 368, 381, 393, 419 377 existence, 351 Cauchy’s rigidity theorem, 43, 143, 213, 279, 339, uniqueness, 350 341, 342, 345, 348–350, 354, 403 algebraic motion, 107, 111 Cauchy—Steinitz lemma, 72, 342 algebraic set, 39, 44 chain algebraic variety, 27 4D, 92, 93, 437 alpha helix, 151, 157, 158 abstract, 65, 149, 153, 158 Amato, Nancy, 157 convex, 143, 145 amino acid, 158 cutting, xi, 91, 123 amino acid residue, 14, 148, 151 equilateral, see -
Memory-Constrained Algorithms for Simple Polygons∗
Memory-Constrained Algorithms for Simple Polygons∗ Tetsuo Asanoy Kevin Buchinz Maike Buchinz Matias Kormanx Wolfgang Mulzer{ G¨unter Rote{ Andr´eSchulzk November 17, 2018 Abstract A constant-work-space algorithm has read-only access to an input array and may use only O(1) additional words of O(log n) bits, where n is the input size. We show how to triangulate a plane straight-line graph with n vertices in O(n2) time and constant work- space. We also consider the problem of preprocessing a simple polygon P for shortest path queries, where P is given by the ordered sequence of its n vertices. For this, we relax the space constraint to allow s words of work-space. After quadratic preprocessing, the shortest path between any two points inside P can be found in O(n2=s) time. 1 Introduction In algorithm development and computer technology, we observe two opposing trends: on the one hand, there are vast amounts of computational resources at our fingertips. Alas, this often leads to bloated software that is written without regard to resources and efficiency. On the other hand, we see a proliferation of specialized tiny devices that have a limited supply of memory or power. Software that is oblivious to space resources is not suitable for such a setting. Moreover, even if a small device features a fairly large memory, it may still be preferable to limit the number of write operations. For one, writing to flash memory is slow and costly, and it also reduces the lifetime of the memory. Furthermore, if the input is stored on a removable medium, write-access may be limited for technical or security reasons. -
Straight Skeletons by Means of Voronoi Diagrams Under Polyhedral Distance Functions
CCCG 2014, Halifax, Nova Scotia, August 11{13, 2014 Straight Skeletons by Means of Voronoi Diagrams Under Polyhedral Distance Functions Stefan Huber∗ Oswin Aichholzery Thomas Hackly Birgit Vogtenhubery Abstract work of Klein [10]. In particular, the straight skeleton does not constitute a Voronoi diagram under some ap- We consider the question under which circumstances the propriate distance function. In this sense, straight skele- straight skeleton and the Voronoi diagram of a given in- tons and Voronoi diagrams are in general fundamentally put shape coincide. More precisely, we investigate con- different. For instance, computing straight skeletons of vex distance functions that stem from centrally symmet- polygons with holes is P -complete [9], but computing ric convex polyhedra as unit balls and derive sufficient Voronoi diagrams is not. Straight skeletons can change and necessary conditions for input shapes in order to ob- discontinuously when input vertices are moved [6], but tain identical straight skeletons and Voronoi diagrams the Voronoi diagram does not. with respect to this distance function. Under these circumstances, it is most interesting that This allows us to present a new approach for general- for rectilinear input in general position (that is, a rec- izing straight skeletons by means of Voronoi diagrams, tilinear polygon where no two edges are collinear) the so that the straight skeleton changes continuously when straight skeleton and the Voronoi diagram under L1- vertices of the input shape are dislocated, that is, no dis- metric indeed coincide [2]. Barequet et al. [5] carried continuous changes as in the Euclidean straight skeleton over this fact to rectilinear polyhedra in three-space.