DOCSLIB.ORG

Robot Online Algorithms in Computational Geometry: Searching and Exploration in the Plane

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Robot Online Algorithms in Computational Geometry: Searching and Exploration in the Plane Robot Online Algorithms in Computational Geometry: Searching and Exploration in the Plane Subir Kumar Ghosh1 Department of Computer Science School of Mathematical Sciences Ramakrishna Mission Vivekananda University Belur Math, Howrah 711202, West Bengal, India [email protected] 1Ex-Professor, School of Technology & Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 11. Mapping polygons using mobile agents. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. Overview of the Lecture 1. O↵-line and online algorithms. 2. Efficiency of online algorithms. 3. Searching for a target on a line. 4. Searching for a target in an unknown region. 5. Continuous and discrete visibility. 6. Searching for a target in an unknown street. 7. Searching for a target in an unknown star-shaped polygon. 8. Exploring an unknown polygon: Continuous visibility. 9. Exploring an unknown polygon: Discrete visibility. 10. Exploring an unknown polygon: Bounded visibility. 11. Mapping polygons using mobile agents. Author's personal copy COMPUTER SCIENCE REVIEW 4(2010)189–201 available at www.sciencedirect.com journal homepage: www.elsevier.com/locate/cosrev Survey Online algorithms for searching and exploration in the plane$ a, b Subir Kumar Ghosh ⇤, Rolf Klein a School of Computer Science, Tata Institute of Fundamental Research, Mumbai 400005, India b Institute of Computer Science I, University of Bonn, Bonn 53117, Germany ARTICLE INFO ABSTRACT Article history: In this paper, we survey online algorithms in computational geometry that have been Received 15 February 2010 designed for mobile robots for searching a target and for exploring a region in the plane. Received in revised form c 2010 Elsevier Inc. All rights reserved. 6 May 2010 Accepted 26 May 2010 Keywords: Online algorithms Motion planning Competitive ratio Target searching Exploration Approximation algorithms Randomized algorithm Computational geometry Shortest path Minimum link path Visibility polygons Watchman route 1. Background situations, it is expected that the robot follows the Euclidean shortest path from s to t inside R. In some other situations, Let s and t be two points in a region R (bounded or unbounded) the robot may be asked to follow a path from s to t inside R in the plane (see Fig. 1). Consider a point robot at s that is which has the minimum number of turns or links. There are searching for the point t in R. If the robot has the complete known efficient sequential algorithms for computing these geometric information (or map) of R and also knows the types of paths from a starting point s to the target point t exact location of t, then the robot can choose a path inside inside a known region R in the plane [1]. Thus, the robot can R to move from s to t. The choice of a path depends compute an optimal path, depending upon the optimization on optimization criteria for the given problem. In many criteria, using its on-board computer system and then follow $ A part of the work was done when the first author visited Institute of Computer Science I, University of Bonn, Bonn 53117, Germany during April–May, 2009. ⇤ Corresponding author. E-mail addresses: [email protected] (S.K. Ghosh), [email protected] (R. Klein). 1574-0137/$ - see front matter c 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.cosrev.2010.05.001 I If the robot has the complete geometric information (or map) of R and also knows the exact location of t,thentherobotcanchooseapath inside R to move from s to t. I In many situations, it is expected that the robot follows the Euclidean shortest path from s to t inside R. I In some situation, the robot may be asked to follow a minimum link (or, turn) path from s to t inside R. O✏ine Algorithms t s3 u s2 s1 R s I Starting from s, a point robot is searching for the point t in R. I In many situations, it is expected that the robot follows the Euclidean shortest path from s to t inside R. I In some situation, the robot may be asked to follow a minimum link (or, turn) path from s to t inside R. O✏ine Algorithms t s3 u s2 s1 R s I Starting from s, a point robot is searching for the point t in R. I If the robot has the complete geometric information (or map) of R and also knows the exact location of t,thentherobotcanchooseapath inside R to move from s to t. I In some situation, the robot may be asked to follow a minimum link (or, turn) path from s to t inside R. O✏ine Algorithms t s3 u s2 s1 R s I Starting from s, a point robot is searching for the point t in R.
Load more

Recommended publications
  • A Visibility-Based Approach to Computing Non-Deterministic
    Article The International Journal of Robotics Research 1–16 A visibility-based approach to computing Ó The Author(s) 2021 Article reuse guidelines: non-deterministic bouncing strategies sagepub.com/journals-permissions DOI: 10.1177/0278364921992788 journals.sagepub.com/home/ijr Alexandra Q Nilles1 , Yingying Ren1, Israel Becerra2,3 and Steven M LaValle1,4 Abstract Inspired by motion patterns of some commercially available mobile robots, we investigate the power of robots that move forward in straight lines until colliding with an environment boundary, at which point they can rotate in place and move forward again; we visualize this as the robot ‘‘bouncing’’ off boundaries. We define bounce rules governing how the robot should reorient after reaching a boundary, such as reorienting relative to its heading prior to collision, or relative to the normal of the boundary. We then generate plans as sequences of rules, using the bounce visibility graph generated from a polygonal environment definition, while assuming we have unavoidable non-determinism in our actuation. Our planner can be queried to determine the feasibility of tasks such as reaching goal sets and patrolling (repeatedly visiting a sequence of goals). If the task is found feasible, the planner provides a sequence of non-deterministic interaction rules, which also provide information on how precisely the robot must execute the plan to succeed. We also show how to com- pute stable cyclic trajectories and use these to limit uncertainty in the robot’s position. Keywords Underactuated robots, dynamics, motion control, motion planning 1. Introduction approach is powerful and well-suited to dynamic environ- ments, but also resource-intensive in terms of energy, Mobile robots have rolled smoothly into our everyday computation, and storage space.
    [Show full text]
  • On Visibility Problems in the Plane – Solving Minimum Vertex Guard Problems by Successive Approximations ?
    On Visibility Problems in the Plane – Solving Minimum Vertex Guard Problems by Successive Approximations ? Ana Paula Tomas´ 1, Antonio´ Leslie Bajuelos2 and Fabio´ Marques3 1 DCC-FC & LIACC, University of Porto, Portugal [email protected] 2 Dept. of Mathematics & CEOC - Center for Research in Optimization and Control, University of Aveiro, Portugal [email protected] 3 School of Technology and Management, University of Aveiro, Portugal [email protected] Abstract. We address the problem of stationing guards in vertices of a simple polygon in such a way that the whole polygon is guarded and the number of guards is minimum. It is known that this is an NP-hard Art Gallery Problem with relevant practical applications. In this paper we present an approximation method that solves the problem by successive approximations, which we intro- duced in [21]. We report on some results of its experimental evaluation and des- cribe two algorithms for characterizing visibility from a point, that we designed for its implementation. 1 Introduction The classical Art Gallery problem for a polygon P is to find a minimum set of points G in P such that every point of P is visible from some point of G. We address MINIMUM VERTEX GUARD in which the set of guards G is a subset of the vertices of P and each guard has 2π range unlimited visibility. This is an NP-hard combinatorial problem both for arbitrary and orthogonal polygons [13, 17]. Orthogonal (i.e. rectilinear) polygons are interesting for they may be seen as abstractions of art galleries, for instance.
    [Show full text]
  • Efficient Computation of Visibility Polygons
    EuroCG 2014, Ein-Gedi, Israel, March 3{5, 2014 Efficient Computation of Visibility Polygons Francisc Bungiu∗ Michael Hemmery John Hershbergerz Kan Huangx Alexander Kr¨ollery Abstract a subroutine in algorithms for other problems, most prominently in the context of the Art Gallery Prob- Determining visibility in planar polygons and ar- lem [16]. In experimental work on this problem [15] rangements is an important subroutine for many al- we have identified visibility computations as having a gorithms in computational geometry. In this paper, substantial impact on overall computation times, even we report on new implementations, and correspond- though it is a low-order polynomial-time subroutine in ing experimental evaluations, for two established and an algorithm solving an NP-hard problem. Therefore one novel algorithm for computing visibility polygons. it is of enormous interest to have efficient implemen- These algorithms will be released to the public shortly, tations of visibility polygon algorithms available. as a new package for the Computational Geometry CGAL, the Computational Geometry Algorithms Algorithms Library (CGAL). Library [5], contains a large number of algorithms and data structures, but unfortunately not for the 1 Introduction computation of visibility polygons. We present im- plementations for three algorithms, which form the Visibility is a basic concept in computational geome- basis for an upcoming new CGAL package for visi- try. For a polygon P R2, we say that a point p P bility. Two of these are efficient implementations for ⊂ 2 is visible from q P if the line segment pq P . The standard O(n)- and O(n log n)-time algorithms from 2 ⊆ points that are visible from q form the visibility re- the literature.
    [Show full text]
  • Parallel Geometric Algorithms. Fenglien Lee Louisiana State University and Agricultural & Mechanical College
    Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1992 Parallel Geometric Algorithms. Fenglien Lee Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Lee, Fenglien, "Parallel Geometric Algorithms." (1992). LSU Historical Dissertations and Theses. 5391. https://digitalcommons.lsu.edu/gradschool_disstheses/5391 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book.
    [Show full text]
  • A Time-Space Trade-Off for Computing the K-Visibility Region of a Point In
    A Time-Space Trade-off for Computing the k-Visibility Region of a Point in a Polygon∗ Yeganeh Bahooy Bahareh Banyassadyz Prosenjit K. Bosex Stephane Durochery Wolfgang Mulzerz Abstract Let P be a simple polygon with n vertices, and let q 2 P be a point in P . Let k 2 f0; : : : ; n − 1g. A point p 2 P is k-visible from q if and only if the line segment pq crosses the boundary of P at most k times. The k-visibility region of q in P is the set of all points that are k-visible from q. We study the problem of computing the k-visibility region in the limited workspace model, where the input resides in a random-access read-only memory of O(n) words, each with Ω(log n) bits. The algorithm can read and write O(s) additional words of workspace, where s 2 N is a parameter of the model. The output is written to a write-only stream. Given a simple polygon P with n vertices and a point q 2 P , we present an algorithm that reports the k-visibility region of q in P in O(cn=s + c log s + minfdk=sen; n log logs ng) expected time using O(s) words of workspace. Here, c 2 f1; : : : ; ng is the number of critical vertices of P for q where the k-visibility region of q may change. We generalize this result for polygons with holes and for sets of non-crossing line segments. Keywords: Limited workspace model, k-visibility region, Time-space trade-off 1 Introduction Memory constraints on mobile devices and distributed sensors have led to an increasing focus on algorithms that use their memory efficiently.
    [Show full text]
  • Optimizing Terrestrial Laser Scanning Measurement Set-Up
    OPTIMIZING TERRESTRIAL LASER SCANNING MEASUREMENT SET-UP Sylvie Soudarissanane and Roderik Lindenbergh Remote Sensing Department (RS)) Delft University of Technology Kluyverweg 1, 2629 HS Delft, The Netherlands (S.S.Soudarissanane, R.C.Lindenbergh)@tudelft.nl http://www.lr.tudelft.nl/rs Commission WG V/3 KEY WORDS: Laser scanning, point cloud, error, noise level, accuracy, optimal stand-point ABSTRACT: One of the main applications of the terrestrial laser scanner is the visualization, modeling and monitoring of man-made structures like buildings. Especially surveying applications require on one hand a quickly obtainable, high resolution point cloud but also need observations with a known and well described quality. To obtain a 3D point cloud, the scene is scanned from different positions around the considered object. The scanning geometry plays an important role in the quality of the resulting point cloud. The ideal set-up for scanning a surface of an object is to position the laser scanner in such a way that the laser beam is near perpendicular to the surface. Due to scanning conditions, such an ideal set-up is in practice not possible. The different incidence angles and ranges of the laser beam on the surface result in 3D points of varying quality. The stand-point of the scanner that gives the best accuracy is generally not known. Using an optimal stand-point of the laser scanner on a scene will improve the quality of individual point measurements and results in a more uniform registered point cloud. The design of an optimum measurement setup is defined such that the optimum stand-points are identified to fulfill predefined quality requirements and to ensure a complete spatial coverage.
    [Show full text]
  • Visibility Polygon Traversal Algorithm
    Visibility polygon traversal algorithm Izaki, Åsmund 1 Derix, Christian 1 Keywords: visibility in polygon with holes; query-based algorithm; spatial analysis Abstract The determination of visibility relations within polygonal environments has applications in many different fields; In the current context it is primarily investigated as a basis for spatial architectural analysis, and as a design driver for architectural design from a user and occupant perspective, but it is equally applicable to a wide range of engineering problems and it’s a well- established topic in computational geometry. We introduce a new query-based algorithm for traversing over the visible convex regions of a polygon with holes from any point inside the polygon or from any of its vertices, where each query runs in O(f’ h’ + log n) for a polygon with n vertices, f’ visible convex partitions, and h’ visible holes, with a preprocessing stage that runs in O(n log* n) with O(n) space. The log n component of the query only applies to internal points. The visibility polygon traversal algorithm is applicable to a varied set of visibility problems, including the construction of visibility polygons (isovists), and visibility graphs of polygon vertices and/or points inside the polygon. The algorithmic findings are linked to spatial architectural analysis by representing the regions of architectural or urban plans that are permeable or visually open as polygons with holes. Two applications of the algorithm are presented, which have enabled more responsive and dynamic user interactions with architectural plans. These consist of an interactive isovist tool and a tool for calculating shortest paths, flows and distances within a multi-storey layout.
    [Show full text]
  • 30 POLYGONS Joseph O’Rourke, Subhash Suri, and Csaba D
    30 POLYGONS Joseph O'Rourke, Subhash Suri, and Csaba D. T´oth INTRODUCTION Polygons are one of the fundamental building blocks in geometric modeling, and they are used to represent a wide variety of shapes and figures in computer graph- ics, vision, pattern recognition, robotics, and other computational fields. By a polygon we mean a region of the plane enclosed by a simple cycle of straight line segments; a simple cycle means that nonadjacent segments do not intersect and two adjacent segments intersect only at their common endpoint. This chapter de- scribes a collection of results on polygons with both combinatorial and algorithmic flavors. After classifying polygons in the opening section, Section 30.2 looks at sim- ple polygonizations, Section 30.3 covers polygon decomposition, and Section 30.4 polygon intersection. Sections 30.5 addresses polygon containment problems and Section 30.6 touches upon a few miscellaneous problems and results. 30.1 POLYGON CLASSIFICATION Polygons can be classified in several different ways depending on their domain of application. In chip-masking applications, for instance, the most commonly used polygons have their sides parallel to the coordinate axes. GLOSSARY Simple polygon: A closed region of the plane enclosed by a simple cycle of straight line segments. Convex polygon: The line segment joining any two points of the polygon lies within the polygon. Monotone polygon: Any line orthogonal to the direction of monotonicity inter- sects the polygon in a single connected piece. Star-shaped polygon: The entire polygon is visible from some point inside the polygon. Orthogonal polygon: A polygon with sides parallel to the (orthogonal) coordi- nate axes.
    [Show full text]
  • Online Algorithms with Discrete Visibility
    Online Algorithms with Discrete Visibility Exploring Unknown Polygonal Environments he context of this work is the exploration of algorithms that could compute visibility polygons are time unknown polygonal environments with obstacles. consuming. The computing limitations suggest that it may Both the outer boundary and the boundaries of not be practically feasible to continuously compute the visi- obstacles are piecewise linear. The bound- bility polygon along the robot’s trajectory. Sec- aries can be nonconvex. The exploration ond, for good visibility, the robot’s camera will Tproblem can be motivated by the following typically be mounted on a mast. Such devi- application. Imagine that a robot has to ces vibrate during the robot’s movement, explore the interior of a collapsed build- and, hence, for good precision (which ing, which has crumbled due to an is required to compute an accurate earthquake, to search for human sur- visibility polygon) the camera must vivors. It is clearly impossible to have be stationary at each view. There- a knowledge of the building’s inte- fore, it seems feasible to compute rior geometry prior to the explo- visibility polygons only at a ration. Thus, the robot must be discrete number of points. Hence, able to see, with its onboard in this article, we assume that the vision sensors, all points in the visibility polygon is computed at building’s interior while follow- a discrete number of points. ing its exploration path. In this Although the earlier discus- way, no potential survivors will sion suggests that a robot can be missed by the exploring only compute visibility polygons robot.
    [Show full text]
  • Generating Kernel Aware Polygons
    UNLV Theses, Dissertations, Professional Papers, and Capstones May 2019 Generating Kernel Aware Polygons Bibek Subedi Follow this and additional works at: https://digitalscholarship.unlv.edu/thesesdissertations Part of the Computer Sciences Commons Repository Citation Subedi, Bibek, "Generating Kernel Aware Polygons" (2019). UNLV Theses, Dissertations, Professional Papers, and Capstones. 3684. http://dx.doi.org/10.34917/15778552 This Thesis is protected by copyright and/or related rights. It has been brought to you by Digital Scholarship@UNLV with permission from the rights-holder(s). You are free to use this Thesis in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s) directly, unless additional rights are indicated by a Creative Commons license in the record and/ or on the work itself. This Thesis has been accepted for inclusion in UNLV Theses, Dissertations, Professional Papers, and Capstones by an authorized administrator of Digital Scholarship@UNLV. For more information, please contact [email protected]. GENERATING KERNEL AWARE POLYGONS By Bibek Subedi Bachelor of Engineering in Computer Engineering Tribhuvan University 2013 A thesis submitted in partial fulfillment of the requirements for the Master of Science in Computer Science Department of Computer Science Howard R. Hughes College of Engineering The Graduate College University of Nevada, Las Vegas May 2019 © Bibek Subedi, 2019 All Rights Reserved Thesis Approval The Graduate College The University of Nevada, Las Vegas April 18, 2019 This thesis prepared by Bibek Subedi entitled Generating Kernel Aware Polygons is approved in partial fulfillment of the requirements for the degree of Master of Science in Computer Science Department of Computer Science Laxmi Gewali, Ph.D.
    [Show full text]
  • Computational Problems in Strong Visibility
    Computational Problems in Strong Visibility Eugene Fink Derick Wood School of Computer Science Department of Computer Science Carnegie Mellon University Hong Kong Univ. of Science and Technology Pittsburgh, PA 15213, USA Clear Water Bay, Kowloon, Hong Kong [email protected] [email protected] http://www.cs.cmu.edu/∼eugene http://www.cs.ust.hk/∼dwood ABSTRACT Strong visibility is a generalization of standard visibility, defined with respect to a fixed set of line orien- tations. We investigate computational properties of this generalized visibility, as well as the related notion of strong convexity. In particular, we describe algorithms for the following tasks: 1. Testing the strong visibility of two points in a polygon and the strong convexity of a polygon. 2. Finding the strong convex hull of a point set and that of a simple polygon. 3. Constructing the strong kernel of a polygon. 4. Identifying the set of points that are strongly visible from a given point. Keywords: computational geometry, generalized visibility and convexity, restricted orientations. 1 INTRODUCTION The study of nontraditional notions of visibility is a fruitful branch of computational geometry, which has found applications in VLSI design, computer graphics, motion planning, and other areas. Researchers have investigated multiple variations and generalizations of standard visibility and convexity.1,2,6,11{13,15,18 Rawlins has introduced the notions of strong visibility and convexity as a part of his research on restricted- orientation geometry.15 These notions are defined with respect to a fixed set of line orientations. They are stronger than standard visibility and convexity, hence the name.
    [Show full text]
  • Faculty of Science Solving the Art Gallery Problem in Iterations
    Faculty of Science Dept. of Information and Computing Sciences Solving the Art Gallery Problem in Iterations Author: Simon Hengeveld 5500117 First supervisor: dr. T. Miltzow Second examiner: Prof. dr. M.J. van Kreveld Acknowledgements Firstly, I would like to thank my supervisor Tillmann Miltzow for introducing me to this interesting project. As the project was close to him, he was very interested in and positive about my progression. This alleviated the usual stress that comes with a master's thesis, making the process a lot less tedious and much more enjoyable. Additionally, I am grateful to my second examiner Mark van Kreveld for his valuable feedback and knowledge of the thesis process. I would also like to thank my housemate and friend Christopher Bouma for letting me use his computer to run several experiments. Furthermore, I am thankful to my parents for always being available to offer good advice concerning difficult choices. Finally, special thanks go to my girlfriend Sofia Rosero Abad for bearing with me through the last stressful weeks and helping me make the graphs and figures look attractive. Abstract The Art Gallery problem is a famous problem in the field of Computational Geometry. The variant of the problem discussed in this thesis is defined as follows: given a polygon (possibly with holes) P , the goal is to find the smallest set of point guards G ⊂ P such that every point p 2 P is at least seen by one of the guards g 2 G. A guard sees a point within P if the segment between the guard and the point is contained in P .
    [Show full text]