pursuit curve

A pursuit curve is the path an object takes when chasing after another object in the most effective way. Pursuit curves can arise in a variety of situations, for example when a fox is chasing after a gazelle or a heat-seeking missile is homing in on a moving target. Suppose that four ants are at the corners of a square. They start to crawl clockwise at a constant rate, each moving toward its neighbor. At any instant, they mark the corners of a square. As the ants get closer to the original square's center, the new square they define rotates and diminishes in size. In reaching the center, each ant travels on a logarithmic spiral with a length equal to the side of the original square. Superimposed snapshots of the ants' progress add up to an intriguing pattern.