Abstract

What do submarine attacks, ant trails, and dating have in common? Not much, except that they are all instances of pursuit and evasion problems and all submit to elegant mathematical treatments. The mathematics involved in such problems is varied and interesting in its own right, but the applications breathe life into the mathematics and invite wider engagement—as the intense interest of the military in such problems, especially during wartime, demonstrates. Consider the problem of a submarine commander about to fire on an enemy warship. The enemy warship, of course, refuses to co operate, and the warship keeps up its course across the open sea. The submarine commander needs a strategy to ensure that the torpedoes arrive where the warship will be by the time the torpedoes arrive. How does the commander do this, without knowing the warship’s course? After all, there is no guarantee that the warship will travel in a straight line or at a constant speed. One way for the submarine commander to proceed is to keep the torpedo always aimed at the ship, instantaneously adjusting its direction to changes in the warship’s position. If this is done, the torpedo will (typically) trace out an arc and (typically) strike the ship from behind. This is called a pure pursuit strategy and is important in many military applications (such as Second World War air battles).