Pursuit Curves
Place N agents at the vertices of a regular polygon. Each one chases the next — agent 1 pursues agent 2, agent 2 pursues agent 3, and so on around the ring. No strategy, no prediction, just constant bearing toward the target. The paths they trace are pursuit curves: logarithmic spirals that converge to the center with a symmetry that belies the simplicity of the rule.
Bouguer, 1732 · the “four bugs” problem · d = L / (1 − cos(2π/n))
What pursuit curves are
A pursuit curve is the path traced by an agent that always moves directly toward a moving target. At every instant, the pursuer reorients to face the prey and advances at a constant speed. There is no prediction, no interception strategy, no game theory — just the simplest possible chase rule: move toward where the target is right now. The result is a curve, not a straight line, because the target is moving too. As the target shifts, the pursuer continually adjusts its heading, and the accumulated corrections produce a smooth, curving trajectory. When multiple agents are arranged symmetrically and each chases the next in a ring, the curves they trace are pursuit curves with rotational symmetry — and they are unexpectedly beautiful.
The four bugs problem
The classic formulation is attributed to Pierre Bouguer in 1732, though the version most people encounter is the “four bugs” problem: four insects sit at the corners of a square, and each walks directly toward the next one (clockwise or counterclockwise). Because of symmetry, all four bugs are always at the corners of a square that is simultaneously shrinking and rotating. The bugs spiral inward along logarithmic curves and meet at the center, having each walked a distance exactly equal to the side length of the original square. For a general regular polygon with n sides of length L, each pursuer walks a total distance of L / (1 − cos(2π/n)) before all agents converge. For a square (n = 4), this simplifies to L. For a triangle (n = 3), it is 2L / 3. As n increases, the total distance grows: agents on a polygon with many sides are nearly chasing the agent directly ahead of them, which is moving nearly parallel, so convergence takes longer.
Why logarithmic spirals
When agents start at the vertices of a regular polygon, symmetry is preserved throughout the pursuit. At every instant, the agents occupy the vertices of a regular n-gon that has rotated and contracted. The center of the polygon is a fixed point, and each agent’s distance to the center decreases exponentially while the polygon rotates at a constant angular rate. This combination — exponential radial contraction plus uniform angular velocity — is the defining property of a logarithmic (equiangular) spiral. The spiral makes a constant angle with every radius vector from the center, and that angle depends only on n: it equals π/2 − π/n. For a square, the angle is 45°. For a triangle, 30°. The logarithmic spiral appears everywhere in nature — nautilus shells, hurricane arms, spiral galaxies — and here it emerges from the simplest possible rule of pursuit.
The general pursuit differential equation
For a single pursuer chasing a single prey, the pursuit curve satisfies a second-order ordinary differential equation. If the prey moves along a known path and the pursuer always heads directly toward the prey’s current position at speed v, the constraint is that the tangent to the pursuer’s path always points at the prey. This leads to: y′′ = (ds/dx) · (1/k), where s is arc length along the pursuer’s path and k is the ratio of pursuer speed to prey speed. For the special case where the prey moves along a straight line, the equation can be solved in closed form, yielding curves that are either algebraic or transcendental depending on the speed ratio. When the ratio is exactly 1, the pursuer asymptotically approaches but never catches a prey moving in a straight line. When the ratio exceeds 1, capture occurs in finite time.
The capture ratio
The slider labeled “capture ratio” controls the speed of the pursuer relative to the effective evasion speed. When the ratio equals 1 (the classic case), each agent moves at the same speed and the polygon shrinks to a point. When the ratio is greater than 1, the pursuer is faster than the prey’s effective escape, and capture happens more quickly — the spirals tighten. When the ratio is less than 1, the pursuer is too slow to ever converge, and the curves spiral outward or trace open paths that never close. This transition — from convergence to divergence — is sharp and depends on the geometry of the polygon as well as the speed ratio.
Connections to pursuit in the real world
Pursuit curves are not just a mathematical curiosity. Proportional navigation — a guidance law used by missiles — is a refinement of the pure pursuit rule. Instead of always pointing at the target, the missile steers to maintain a constant bearing angle, which produces a more efficient intercept. But the analysis begins with pure pursuit as the baseline case. In biology, many predators use approximate pursuit strategies: dragonflies intercept prey using a pursuit-like algorithm implemented in neural circuits, and wolves chase prey on curves that resemble mathematical pursuit paths. In robotics, pursuit algorithms are used for formation control in multi-robot swarms: each robot follows the one ahead, and the resulting trajectories converge to desired formations. The pursuit curve is one of the oldest problems in differential equations, and it remains one of the most directly applicable.