The Oldest Navigation Method
Before GPS, before radio, before chronometers, sailors navigated by dead reckoning — estimating position from course steered and distance sailed. Columbus crossed the Atlantic using DR, checking his position against the stars when skies were clear. The method is beautifully simple: if you know where you started, which direction you went, and how far, you know where you are. Its weakness is equally simple: errors accumulate with every mile.
Along-Track and Cross-Track Errors
Dead reckoning errors decompose into two components. Along-track error arises from imperfect speed measurement — if the speedometer reads 3% high, you think you have traveled 515 m when you have actually traveled 500 m. Cross-track error arises from heading error — if your compass is off by 2°, you drift sideways by d·sin(2°) ≈ 35 m per kilometer. The simulation shows both error components growing as the vehicle moves along its path.
Error Ellipse Growth
The total position uncertainty forms an ellipse that elongates over time. Cross-track error typically dominates because heading error multiplies by the full distance traveled, while speed error only affects the along-track component proportionally. After 10 km with 2° heading error and 3% speed error, the cross-track error (~350 m) dwarfs the along-track error (~300 m). The simulation draws this growing uncertainty ellipse around the estimated position.
Modern Dead Reckoning
Today, dead reckoning fills the gaps between GPS fixes. Automotive navigation systems use wheel odometry (speed) and a gyroscope (heading) to maintain positioning through tunnels and parking garages. Pedestrian dead reckoning uses smartphone accelerometers to count steps and estimate distance. Robot vacuum cleaners use wheel encoders for room mapping. In every case, the fundamental principle is the same as Columbus used — but the sensors and update algorithms have improved by orders of magnitude.