Chase and Escape: Pursuit Problems

Jolly-Roger From cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.

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In the swashbuckling days of the buccaneers, ships laden with treasures were targets for pirates. Once spotted, they became prey in a life and death chase. Could they reach safety before they were overhauled? Again, a pursuit problem is involved and basic mathematical reasoning may be the means of escape.

The mathematical study of pursuit has a long history. The first comprehensive treatment was that of Pierre Bouguer around 1730. Bouguer considered a pirate ship in pursuit of a fleeing merchantman, where the chasing ship always sails towards its target. If the pirate ship is faster, capture is inevitable unless the merchant ship can reach a safe haven in time.

Cyclic Pursuit Problems

One of the classical cases of pursuit is the N-bug problem (it has several other names). It has a long history and has been discussed by many authors. There are N bugs at the vertices of a regular N-gon. Each moves directly towards its immediate clockwise neighbour at the same rate. How far do they travel and how soon do they all meet at the centre?

N-Bugs-1 — Four bugs initially at the corners of the square ABCD. Each moves at a fixed rate towards its `clockwise’ neighbour.

The case N=4 is illustrated. By symmetry, the 4 bugs are always at the corners of a square, the size of which decreases with time. It is a fairly easy exercise in elementary calculus to show that the paths of the bugs are logarithmic spirals

$\displaystyle r(\theta) = r_0 \exp[-\theta \tan(\pi/N) ]$

For the 4-bug problem, this takes the simple form ${r(\theta) = r_0 \exp(-\theta)}$ . The time and distance questions can be answered using this formula. However, they are easily solved by simpler reasoning. Suppose that each side of the N-gon is of length ${L}$ . The external angle of the N-gon is ${2\pi/N}$ . Thus, if each bug goes at a speed ${V}$ , the relative speed, or closing speed, of two neighbouring bugs is ${V[1-\cos(2\pi/N)]}$ . For ${N=4}$ , this is just ${V}$ (the paths of the two bugs are orthogonal), so the distance travelled is ${s = L}$ and the time to meet is ${T = L / V}$ .

Steinhaus (1950) discusses the 4-bug problem, points out that the spirals always cut the diagonals of the square at an angle of ${45^\circ}$ , and asks why this is the case. You may wish to reflect on this.

Pretty Graphics

Since the four bugs are always at the corners of a square, we can illustrate the solution by plotting these squares at different times. They are shown below, and the spirals are evident as an `emergent phenomenon’.

N-Bugs-2 — Four bugs at the corners of a series of squares. The emergence of the spiral paths is clear.

Three bugs initially at the vertices of an equilateral triangle remain in this configuration as the scale decreases. The diminishing triangles are shown below, and an explicit plot of the three spirals is also shown.

N-Bugs-36 — Left: Three bugs initially at the vertices of an equilateral triangle. Right: Three spiral paths for the 3 bugs.

Problems in which the predator always moves directly towards its prey are called `pure pursuit problems’. However, this is not generally an optimal strategy. A submarine firing a torpedo at a moving battleship must calculate an intercept course, where the movement of the target is used to determine a point of impact.

With actively guided torpedoes, in-course computations are continually refreshed as the target undertakes evasive actions. It is a regrettable reality that the applications of mathematics are not always in the interests of peace.

Sources

${\bullet}$ Nahin, Paul J, (2007). Chases and Escapes: the Mathematics of Pursuit and Evasion. Princeton Univ. Press, 253pp.

${\bullet}$ Steinhaus, Hugo, 1950: Mathematical Snapshots. Oxford Univ. Press, 311pp.

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