DOCSLIB.ORG

Computation and Complexity of Visibility in Geometric Environments

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Computation and Complexity of Visibility in Geometric Environments Computation and complexity of visibility in geometric environments Computation and complexity of visibility in geometric environments Berekenen en complexiteit van zichtbaarheid in geometrische omgevingen (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. J.C. Stoof, ingevolge het besluit van het college voor promoties in het openbaar te verdedigen op woensdag 9 april 2008 des middags te 12.45 uur door Esther Jantje Moet geboren op 28 februari 1980 te Dirksland Promotor: Prof.dr. M. H. Overmars Co-promotor: Dr. M. J. van Kreveld Contents iii Contents 1 Introduction 1 1.1 Motivation . .2 1.1.1 Visibility in robotics . .2 1.1.2 Visibility in GIS . .3 1.1.3 Visibility in computer graphics . .4 1.2 Visibility problems in computational geometry . .5 1.2.1 Modeling the world . .6 1.2.2 Modeling visibility . .7 1.3 Types of problems . .9 1.3.1 Guarding problems . 10 1.3.2 Computing what is visible . 11 1.3.3 Analytical data structures . 12 1.4 Contributions of this thesis . 13 2 Guarding art galleries by guarding witnesses 15 2.1 Introduction . 16 2.2 Preliminaries . 17 2.3 Examples . 21 2.4 Finite witness sets . 23 2.5 Minimum size witness sets . 28 2.5.1 Reflex-reflex and convex-reflex edges . 28 2.5.2 Convex-convex edges . 29 2.5.3 Placement of witnesses . 29 2.5.4 Size of a minimum size witness set . 30 2.6 Algorithms . 31 2.6.1 Visibility kernel arrangement . 31 2.6.2 Algorithms . 33 2.7 Concluding remarks . 35 3 Region intervisibility in terrains 37 3.1 Introduction . 38 3.2 Preliminaries . 39 3.3 Complete intervisibility . 40 iv Contents 3.3.1 Geometric properties . 40 3.3.2 Algorithm . 42 3.3.3 Running time analysis . 44 3.3.4 3sum-hardness . 45 3.4 Semi-complete intervisibility . 47 3.4.1 Aspect graph approach . 48 3.4.2 Using the edge visibility map . 49 3.5 Partial intervisibility . 50 3.6 Concluding remarks . 51 4 Visibility maps of segments and triangles in 3D 53 4.1 Introduction . 54 4.2 Preliminaries . 56 4.3 The strong visibility map of an edge . 58 4.4 The weak visibility map of an edge . 63 4.4.1 Lower bounds . 63 4.4.2 Upper bounds . 64 4.4.3 Algorithm . 66 4.5 The triangle visibility map . 68 4.5.1 The strong visibility map of a triangle . 68 4.5.2 The weak visibility map of a triangle . 69 4.5.3 Algorithm . 71 4.6 Concluding remarks . 72 5 On realistic terrains 73 5.1 Introduction . 74 5.2 Input model . 76 5.2.1 No subset of assumptions is sufficient . 79 5.2.2 Auxiliary results . 82 5.2.3 Dihedral angles . 84 5.3 The visibility map . 86 5.3.1 Upper bounds . 86 5.3.2 Lower bounds . 88 5.4 The shortest path map and the Voronoi diagram . 90 5.4.1 Bisectors . 90 5.4.2 Shortest path maps . 92 5.4.3 Voronoi diagram . 97 5.5 Concluding remarks . 99 6 Experimental verification of a realistic input model for terrains 101 6.1 Introduction . 102 6.2 Experimental setup . 103 6.2.1 DEM data . 103 6.2.2 TIN generation . 104 6.3 The model . 108 Contents v 6.4 Evaluation of the model parameters . 108 6.4.1 Aspect ratio . 108 6.4.2 Fatness . 109 6.4.3 Edge length ratio . 111 6.4.4 Dihedral angle . 112 6.4.5 Computation . 113 6.4.6 Results . 114 6.5 Conclusions . 117 Screenshots of the terrains . 118 7 Conclusions 127 Acknowledgements 132 Bibliography 133 Samenvatting 147 Index 146 Curriculum vitae 151 vi Contents 1 Chapter 1 Introduction For many research fields, it is clear what you study and why. In medicine, we want to make people as healthy as possible. In biology and physics, we try to understand the physical world around us. In history, we study the past to learn from our mistakes for the future. But for many people, it is not very clear what a computer scientist actually does. One way to describe our daily job is that we try to describe the world around us mathematically, and then solve problems from this real world with the aid of a computer. Problems that are studied by computer scientists usually originate in other areas of research or in applications in business or industry. These problems come to the attention of computer scientists because other people encountered these problems in real life, and were not able to solve them with the techniques that they had at their disposal, or not within a reasonable time frame, or not without a computer. To be able to perform a (theoretical) computer science study, it is necessary to ab- stract from the applications and concentrate on the fundamental mathematical problems that lie underneath. In this thesis, we study visibility problems, which are geometric in nature. This leads us to the field of computational geometry (CG). The main topic of this thesis is visibility amidst objects in two and three dimensional environments. Studying fundamental and theoretical problems does not mean we can forget about the motivation behind what we do. Even the reason for studying a particular funda- mental problem can still come from a real-world application. When we study abstract problems, whose solutions sometimes are merely of theoretical interest, it is important to keep in mind exactly why we study these problems. Therefore, we give an extensive overview of applications and application areas in which visibility plays a role. This motivation covers a large part of this first chapter. After that, we discuss visibility research in computational geometry: previous research and the topics of this thesis, and the relation between these two. 2 Introduction 1.1 Motivation There are three application areas that are viewed as the main sources of the need to study visibility: robotics, GIS, and computer graphics. We discuss these fields in more detail below. This overview is not intended to be complete; in the first place it tries to give a glimpse of the many appearances of visibility. 1.1.1 Visibility in robotics Robot motion planning and robotic surveillance raised numerous different visibility chal- lenges which, once described geometrically, posed many interesting theoretical problems. The very first visibility problem was actually motivated by security issues: where do you need to place cameras, and how many of them, in order to guard your valuables? More details on this so-called Art Gallery Problem are given in Section 1.3.1. The best-known use of visibility in robotics is the use of the visibility graph in exact robot motion planning [117, 120, 146]. The visibility graph is a graph in which each vertex represents a (point) location, and if two locations can see each other, then an edge is drawn between them. Such an edge corresponds to a straight visible path from one location to the other. This path is in fact the shortest path between these two locations. In a simple version of the exact robot motion planning problem, an environment with obstacles and the source and target position for a point robot are given. The algorithm computes the visibility graph of the vertices of the obstacles, then finds a visible visibility graph vertex for both the source and the target position, finds the shortest path in the visibility graph between these two vertices, and then combines all the information to obtain a collision-free path. With the output-sensitive visibility graph construction algorithm by Ghosh and Mount [90], it is possible to find a collision- free path in O(n log n + K) time, where n is the total number of vertices of the objects, and K is the number of edges in the visibility graph. Another important topic in robotics is the exploration of, and navigation in, (par- tially) unknown environments, see [52, 187] and several chapters in [33, 48, 189] for an overview of problems and techniques. Obviously, visibility plays an important role herein, since we must see something to be able to explore it. By examining the view for a robot at different times and positions, the environment becomes known little by little. Exploring such an unknown environment often yields a (2- or 3-D) model that can be used later for other purposes, and this technique is therefore also called map or model building [8, 127, 199]. A strongly related topic is that of visibility-based lo- calization: given the view around a robot, determine where it is located in a known environment [116, 188, 198]. Often, these two problems are tackled at the same time, which is studied in the important subfield in robotics called SLAM: Simultaneous Lo- calization and Mapping [126, 177, 187, 189]. Yet another related topic is motion planning with visibility constraints [118, 184]. Such problems are usually set in dynamic environments. This problem has many vari- ants, but the common characteristic is that both motion and visibility constraints need to be satisfied, and often another criterion, such as path length, can be optimized. This visibility-constrained motion planning has applications in, among others, pursuit- 1.1 Motivation 3 evation problems [85, 95, 144, 167], in which one wants to find and track a (hiding) target and first achieve and then maintain visibility. Many of these problems also have applications in games, in both serious and recreational gaming. Visibility is often modeled theoretically by the existence of a simple line-of-sight.
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]
  • 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.
    [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]