The Clairvoyant Curves Of Pursuit

star shapedLet’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
  • Lead pursuit: Direction of your velocity is always ahead of the target
  • Lag pursuit: Direction of your velocity is always behind the target

There are nice mathematical equations describing these curves of pursuit and they get increasingly complex as the motion of the target becomes more curvy. Nonetheless, it’s fun to read and understand the formulation. If you are interested, you should check this out.

The Shooter, The Hunter and The Coordinator

types of pursuit curvesDifferent types of pursuit curves have different uses. It becomes more pronounced when the target is moving along a curved path. For example, lead pursuit is necessary for all long range sniper kills. This is because it takes time for the bullets to travel through space. Sometimes snipers have to shoot from as far as 2 miles. If the target is that far, you cannot just aim and shoot. All kind of ungodly things come into play here! If you shoot where the target is right now the target will no longer be there when the bullets arrive. Therefore, you must predict the future flight path of the target in order to put the ammunition into the target. This is the essence of aerial combat as well. The ability to predict where the target will be in the future is crucial to all facets of aerial combat.

Pure pursuit is used when you are hunting something down. Your velocity direction is always pointed at the target. If your velocity is more than the target, you are bound to reach the target. Lag pursuit is used when want to strategically place yourself in a chase. You can coordinate with your team to place yourself so that the target has no way to escape.


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