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. In Chases 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 Escapes includes 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.
Some images inside the book are unavailable due to digital copyright restrictions.
About the Author
Table of ContentsPreface to the Paperback Edition xiii
What You Need to Know to Read This Book (and How I Learned What I Needed to Know to Write It) xxvii
Chapter 1. The Classic Pursuit Problem 7
- 1.1 Pierre Bouguer's Pirate Ship Analysis 7
- 1.2 A Modern Twist on Bouguer 17
- 1.3 Before Bouguer: The Tractrix 23
- 1.4 The Myth of Leonardo da Vinci 27
- 1.5 Apollonius Pursuit and Ramchundra's Intercept Problem 29
Chapter 2. Pursuit of (Mostly) Maneuvering Targets 41
- 2.1 Hathaway's Dog-and-Duck Circular Pursuit Problem 41
- 2.2 Computer Solution of Hathaway's Pursuit Problem 52
- 2.3 Velocity and Acceleration Calculations for a Moving Body 64
- 2.4 Houghton's Problem: A Circular Pursuit That Is Solvable in Closed Form 78
- 2.5 Pursuit of Invisible Targets 85
- 2.6 Proportional Navigation 93
Chapter 3. Cyclic Pursuit 106
- 3.1 A Brief History of the n-Bug Problem, and Why It Is of Practical Interest 106
- 3.2 The Symmetrical n-Bug Problem 110
- 3.3 Morley's Nonsymmetrical 3-Bug Problem 116
Chapter 4. Seven Classic Evasion Problems 128
- 4.1 The Lady-in-the-Lake Problem 128
- 4.2 Isaacs's Guarding-the-Target Problem 138
- 4.3 The Hiding Path Problem 143
- 4.4 The Hidden Object Problem: Pursuit and Evasion as a Simple Two-Person, Zero-Sum Game of Attack-and-Defend 156
- 4.5 The Discrete Search Game for a Stationary Evader -- Hunting for Hiding Submarines 168
- 4.6 A Discrete Search Game with a Mobile Evader -- Isaacs's Princess-and-Monster Problem 174
- 4.7 Rado's Lion-and-Man Problem and Besicovitch's Astonishing Solution 181
Solution to the Challenge Problems of Section 1.1 187
Solutions to the Challenge Problems of Section 1.2 190
Solution to the Challenge Problem of Section 1.5 198
Solution to the Challenge Problem of Section 2.2 202
Solution to the Challenge Problem of Section 2.3 209
Solution to the Challenge Problem of Section 2.5 214
Solution to the Challenge Problem of Section 3.2 217
Solution to the Challenge Problem of Section 4.3 219
Solution to the Challenge Problem of Section 4.4 222
Solution to the Challenge Problem of Section 4.7 224
Guelman's Proof 229
What People are Saying About This
This is a well-written and novel book that is comprehensively researched and enthusiastically presented. Nahin offers a very good mixture of elegant math and lively historical interludes. I wasn't aware the topic had such a rich history and wide scope.
Desmond Higham, University of Strathclyde
Nahin provides beautiful applications of calculus, differential equations, and game theory. If you are pursuing an enjoyable collection of mathematical problems and the stories behind them, then your search ends here.
Arthur Benjamin, Harvey Mudd College
This book is a treasure trove of puzzles and an enjoyable read. Nahin's aim is to assemble a varied collection of pursuit-and-evasion problems. Fully worked solutions, from first principles, are presented for each problem. Problems are carefully set in their historical context. I am not aware of another book that covers pursuit-and-evasion problems in anywhere near as much detail as is presented here.
Nick Hobson, creator of the award-winning Web site "Nick's Mathematical Puzzles"
I know of no better way to grasp the basic concepts of calculus than to study pursuit-and-escape problems. Paul Nahin has made a superb survey of the vast field of such problems, from Zeno's paradox of Achilles and the tortoise through the famous four bugs that once made the cover of Scientific American. Not only does he make clear the required differential equations, but he traces each problem's colorful history. No book on the topic could be more definitive or a greater pleasure to read.
Chases and Escapes is a superb treatment of the solutions to a variety of pursuit-evasion problems, some classic and others more contemporary. The content is accessible to undergraduates in mathematics or the physical sciences, with lots of supporting detail included. The author's lively writing style makes for enjoyable reading.
David M. Burton, University of New Hampshire
"I know of no better way to grasp the basic concepts of calculus than to study pursuit-and-escape problems. Paul Nahin has made a superb survey of the vast field of such problems, from Zeno's paradox of Achilles and the tortoise through the famous four bugs that once made the cover of Scientific American. Not only does he make clear the required differential equations, but he traces each problem's colorful history. No book on the topic could be more definitive or a greater pleasure to read."Martin Gardner