Pursuit & Evasion
Simulate pursuit curves where a predator chases prey along mathematical paths. Watch how different pursuit strategies, speed ratios, and prey trajectories produce elegant curves — from simple arcs to logarithmic spirals.
Pursuit Status
How it works
A pursuit curve describes the path traced by a pursuer who always moves directly toward a moving target. The mathematical study of these curves dates to Pierre Bouguer in 1732, who posed the problem: if a pirate ship always sails directly toward a fleeing merchant vessel, what path does it follow? The resulting curve is called the tractrix of pursuit or radiodrome.
In pure pursuit, the predator's velocity vector always points directly at the prey's current position. This is the simplest strategy and the one used by many animals. When the prey moves in a straight line and the predator is faster (speed ratio > 1), the predator always catches the prey along a curved path. When speeds are exactly equal, the predator asymptotically approaches but never catches the prey, tracing a logarithmic spiral. When the predator is slower, capture is impossible along a straight-line prey path.
Proportional navigation is a more sophisticated strategy used by guided missiles: instead of aiming at the target, the pursuer steers to keep the bearing angle to the target changing at a rate proportional to the line-of-sight rotation rate. This produces much straighter intercept paths and is the dominant guidance law for modern missiles. The navigation constant N (typically 3–5) determines how aggressively the pursuer corrects its course.
Constant bearing (or collision course) is the optimal strategy: the pursuer moves so that the bearing to the target remains constant. This is the same principle used by sailors for centuries — if another vessel maintains a constant bearing relative to your own, you are on a collision course. This strategy produces the shortest interception path and is equivalent to leading the target, like a hunter aiming ahead of a running deer.
Pursuit curves appear throughout nature: wolves tracking caribou, peregrine falcons stooping on prey, dragonflies intercepting mosquitoes. Each species uses a pursuit law tuned by evolution to its particular ecological niche. Remarkably, studies show that many predators use strategies closer to proportional navigation than pure pursuit — they do not simply chase, but predict and intercept.