Chases and Escapes

Chases and Escapes: The Mathematics of Pursuit and Evasion

Paul J. Nahin
Copyright Date: 2007
Pages: 272
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  • Book Info
    Chases and Escapes
    Book Description:

    We all played tag when we were kids. What most of us don't realize is that this simple chase game is in fact an application of pursuit theory, and that the same principles of games like tag, dodgeball, and hide-and-seek are also at play in military strategy, high-seas chases by the Coast Guard, and even romantic pursuits. InChases and Escapes, Paul Nahin gives us the first complete history of this fascinating area of mathematics, from its classical analytical beginnings to the present day.

    Drawing on game theory, geometry, linear algebra, target-tracking algorithms, and much more, Nahin also offers an array of challenging puzzles with their historical background and broader applications.Chases and Escapesincludes solutions to all problems and provides computer programs that readers can use for their own cutting-edge analysis.

    Now with a gripping new preface on how the Enola Gay escaped the shock wave from the atomic bomb dropped on Hiroshima, this book will appeal to anyone interested in the mathematics that underlie pursuit and evasion.

    eISBN: 978-1-4008-4206-3
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xii)
  3. Preface to the Paperback Edition
    (pp. xiii-xxvi)
    Paul J. Nahin
  4. What You Need to Know to Read This Book (and How I Learned What I Needed to Know to Write It)
    (pp. xxvii-xxx)
  5. Introduction
    (pp. 1-6)

    “Chases and escapes” are activities that might be said to define human existence, no matter where or when, ranging over the entire spectrum of violence from the romantic pursuit of a future spouse to the military pursuit of a target to be destroyed. Young children are introduced to both chases and escapes when they learn the simple rules of the game of tag: played by at least two, one player is designated asit, who then chases after all the others, who, of course, attempt to escape. When theitplayer manages to catch one of the evaders he or...

  6. Chapter 1 The Classic Pursuit Problem
    (pp. 7-40)

    Modern mathematical pursuit analysis is generally assumed to begin with a problem posed and solved by the French mathematician and hydrographer Pierre Bouguer (1698–1758) in 1732. This general assumption is not quite correct, as I’ll soon elaborate, but Bouguer’s problem is today nevertheless taken as the starting point of pursuit analysis in all modern textbooks, and I’ll do the same here. In his paper, read before the French Academy on January 16, 1732, and published in the Academy’sMémoires de l’Académie Royale des Sciencesin 1735, Bouguer treated the case of a pirate ship pursuing a fleeing merchant vessel,...

  7. Chapter 2 Pursuit of (Mostly) Maneuvering Targets
    (pp. 41-105)

    In chapter 1 we considered only the case of the pursued (the chased one, the fugitive, the evader, the escaper, the target, etc.) as always moving along a straight path. It didn’t take very long after Bouguer’s original 1732 statement of the pursuit problem, however, for analysts to begin to consider more complicated evasion paths (what Bouguer called theligne de fuite). The simplest suchflightorfleeingpath (in terms ofshape) would, for most people, be a circular path, and one finds such paths being discussed as early as 1742. These early extensions of the pursuit problemstatement...

  8. Chapter 3 Cyclic Pursuit
    (pp. 106-127)

    In the May 1877 issue ofNouvelle correspondance mathématiquethe French mathematician Edouard Lucas (1842-1891)formallyposed the following pursuit problem, the first really new innovation in pursuit questions since Bouguer’s original problem:

    Three dogs are placed at the vertices of an equilateral triangle; they run one after the other. What is the curve described by each of them?

    The answer was not long in coming: in the August issue Henri Brocard (mentioned in the previous chapter) stated that, if we suppose the dogs to start at the same time and to run with the same speed, then the pursuit...

  9. Chapter 4 Seven Classic Evasion Problems
    (pp. 128-186)

    This is a clever pursuit-and-evasion problem with the emphasis onevasion. At its most elementary level it became famous decades ago when, like the four-bug cyclic pursuit problem in the previous chapter, it appeared in Martin Gardner’s “Mathematical Games” column inScientific American(November and December 1965). As Gardner presented the problem then to his readers,

    A young lady was vacationing on Circle Lake, a large artificial body of water named for its precisely circular shape. To escape from a man who was pursuing her, she got into a rowboat and rowed to the center of the lake, where a...

  10. Appendix A Solution to the Challenge Problems of Section 1.1
    (pp. 187-189)
  11. Appendix B Solutions to the Challenge Problems of Section 1.2
    (pp. 190-197)
  12. Appendix C Solution to the Challenge Problem of Section 1.5
    (pp. 198-201)
  13. Appendix D Solution to the Challenge Problem of Section 2.2
    (pp. 202-208)
  14. Appendix E Solution to the Challenge Problem of Section 2.3
    (pp. 209-213)
  15. Appendix F Solution to the Challenge Problem of Section 2.5
    (pp. 214-216)
  16. Appendix G Solution to the Challenge Problem of Section 3.2
    (pp. 217-218)
  17. Appendix H Solution to the Challenge Problem of Section 4.3
    (pp. 219-221)
  18. Appendix I Solution to the Challenge Problem of Section 4.4
    (pp. 222-223)
  19. Appendix J Solution to the Challenge Problem of Section 4.7
    (pp. 224-228)
  20. Appendix K Guelman’s Proof
    (pp. 229-234)
  21. Notes
    (pp. 235-244)
  22. Bibliography
    (pp. 245-248)
  23. Acknowledgments
    (pp. 249-250)
    Paul J. Nahin
  24. Index
    (pp. 251-253)
  25. Back Matter
    (pp. 254-254)