As the bee wanders outward it repeatedly measures its own heading and the distance travelled since the last instant, then adds that small step — as a vector — onto a running total. This running total, the path integral, always points back the way it thinks the hive is, even along a route that never went in a straight line.
V_home(t) = -Σ Δs_i · (cos θ_i, sin θ_i)
|V_home| = distance estimate, arg(V_home) = bearing home
error(t) ≈ σ_noise · √N (dead-reckoning error grows with steps N)
- Flight speed — how fast the bee covers ground; faster flights accumulate the same distance in fewer integration steps.
- Sensor noise — how imprecise each step's heading/distance estimate is (compass and odometer error); higher noise makes the home vector drift further from the true displacement.
- Fly to forage — releases the bee on a meandering outbound search; the gold trail is the true flown path, the amber arrow is the bee's live integrated home vector.
- Fly home — once food is found, the bee ignores its own trail entirely and flies the straight line given by the integrated vector — proof the vector, not the route, is what's remembered.
- New round — starts another foraging trip; the learning curve tracks how return error changes as the bee gets more experience with the same terrain.
Real bees combine this odometry with celestial-compass and landmark cues, and mushroom-body circuits refine the estimate with experience — which is why homing accuracy in real colonies improves measurably after a bee's first few foraging trips, just as the learning curve here trends downward.