PURSUIT THEORY
How does chasing work?
Pirate Ships and Pocket Watches
It seems odd that these two items would have much in common, but they have both been used to lay down a theoretical framework for pursuit.
Perrault’s Pocket Watch
Around 1672, physician Claude Perrault posed a then unsolved question about motion to the polymath Gottfried Leibniz. If a watch on a chain is placed on a table, and the end of the chain dragged in a straight line, what path does the watch take? The experiment is easy; the mathematical analytical solution, harder (particularly for the time). We need not worry about this solution, but it’s worth considering that in this case, the watch is pursuing the hand that pulls the chain. However, its speed is variable and dependent on the angle of the chain. The curve made by an object dragged in this manner is called a Tractrix. It’s not important further, it’s just a cool name.
Bouguer’s Pirate Ship
In 1732, mathematician Pierre Bouguer laid down a hypothetical case in which a pirate ship sets to attacking a passing merchant. In this case, the pirate ship is always pointed directly toward the merchant’s. The question proposed, and solved, was: For a given speed, what path does the pirate take towards his quarry? This is closer to the problem encountered in pursuit in real life. Chaser speed is independent of the target speed, unlike in the tractrix. When chasing a fleeing target, the pirate can only catch it when his speed is higher.
In French, the shape made is a courbe de chien and in German it’s a hundekurven. Both mean: the curve of a dog. Both share the image of a dog tailing its owner, our first connection to understanding how animals might complete pursuit tasks
Staying in control
The movements of animals aren’t defined by mathematical hypotheticals. Instead, they’re driven by control. How does an animal achieve the path of Bourguer’s Pirate?