# The Clairvoyant Curves Of Pursuit

Let’s say you see a rogue fighter aircraft on your territory and you want to demolish it. Well, not you per se, but a person who is in charge of these things! Anyway, you send your pilot so that he can chase the aircraft and destroy it. But how will he determine the path of his aircraft so that it takes the shortest path? Bear in mind that the target is moving. If the target moves in a straight line, this is a pretty simple problem. But it almost never happens in real life. Also, if you chase the target but arrive there in a vulnerable position, you might get attacked by the target itself. You need to arrive there in time and also in an offensive position. How do we solve this?

Why do we need pursuit curves?

Pursuit curves are used by fighter pilot to solve the geometry problems associated with combat situations. Any time you are chasing a target, you are applying pursuit curves. An understanding of when to use these things will improve your chances of arriving at the target in an offensive position. In the above example, you can use pursuit curves to determine where the target will be at a future point of time. A missile chasing a ship uses pursuit curves to determine the shortest path. Pursuit curves occur in nature as well. A lion chasing a gazelle uses pursuit curves to determine how it can place itself to pounce on the target when it catches up. Also, pursuit curves are often extended to other domains as well. You don’t need an actual physical target and an actual chase to use pursuit curves. The path of the target can be abstract events too. This is applicable when chasing criminals based on their behavior. You need to understand their path to determine how to pursue them.

What exactly are pursuit curves?

If a prey moves along a known curve, then the path of the predator describes a pursuit curve. A curve of pursuit is a curve constructed by analogy to having a point or points which represents pursuers and pursuees, and the curve of pursuit is the curve traced by the pursuers. With the paths of the pursuer and pursuee parameterized in time, the pursuee is always on the pursuer’s tangent. One particle travels along a specified curve, while a second pursues it, with a motion always directed toward the first. The velocities of the two particles are always in the same ratio. There are mainly three different types of pursuit curves:

• Pure pursuit: Direction of your velocity is always towards the target