Dead Reckoning Formula
Calculate your current position from a known starting point using heading, speed, and elapsed time.
A fundamental navigation technique.
The Formula
Longitude₂ = Longitude₁ + d × sin(θ) / cos(Latitude₁)
Dead reckoning (often abbreviated DR) is one of the oldest and most fundamental navigation techniques. It estimates your current position by starting from a known location and applying your heading and distance traveled. The name likely derives from "deduced reckoning," though sailors have used the term for centuries.
The core idea is straightforward: if you know where you started, which direction you went, and how far you traveled, you can calculate where you are now. Distance is simply speed multiplied by time (d = v × t). The heading angle determines how that distance is split between north-south movement (latitude change) and east-west movement (longitude change).
The cosine of the heading gives the northward component, and the sine gives the eastward component. The longitude formula includes an additional division by cos(Latitude) to account for the fact that lines of longitude converge toward the poles. One degree of longitude covers less distance the farther you are from the equator.
Dead reckoning is inherently imperfect because it accumulates errors over time. Wind, currents, steering inaccuracies, and speed estimation errors all compound with each passing hour. This is why navigators historically combined dead reckoning with celestial observations, and modern navigators use it as a backup to GPS.
Despite its limitations, dead reckoning remains essential in aviation, marine navigation, and robotics. Pilots use it for flight planning and as a fallback if electronic systems fail. Submarines rely on inertial navigation systems that are essentially sophisticated dead reckoning computers, since GPS signals cannot penetrate water.
For short distances on Earth (under a few hundred kilometers), this flat-Earth approximation works well. For longer distances, spherical trigonometry or the Vincenty formulas provide better accuracy.
Variables
| Symbol | Meaning |
|---|---|
| Latitude₁, Longitude₁ | Starting position (in decimal degrees) |
| Latitude₂, Longitude₂ | Estimated new position (in decimal degrees) |
| d | Distance traveled (in degrees; 1° latitude ≈ 111.32 km or 60 nautical miles) |
| θ | Heading or course angle (measured clockwise from true north, in radians) |
| v | Speed (used to compute d = v × t) |
| t | Elapsed time |
Example 1
A ship starts at 40°N, 74°W heading due northeast (045°) at 20 knots for 3 hours. Find the new position.
Distance = 20 knots × 3 hours = 60 nautical miles = 1° of latitude
Convert heading: θ = 45° = 0.7854 radians
Latitude₂ = 40° + 1° × cos(45°) = 40° + 0.707° = 40.707°N
Longitude₂ = -74° + 1° × sin(45°) / cos(40°) = -74° + 0.707° / 0.766 = -74° + 0.923°
New position: 40.707°N, 73.077°W
Example 2
An aircraft at 51.5°N, 0°W (London) flies heading 180° (due south) at 450 knots for 1 hour. Where does it end up?
Distance = 450 nautical miles = 450/60 = 7.5° of latitude
Latitude₂ = 51.5° + 7.5° × cos(180°) = 51.5° + 7.5° × (-1) = 51.5° - 7.5° = 44.0°N
Longitude₂ = 0° + 7.5° × sin(180°) / cos(51.5°) = 0° + 0 = 0°
New position: 44.0°N, 0°W (over southern France, near Bordeaux)
When to Use It
Dead reckoning is used whenever you need to estimate position from movement data alone.
- Marine navigation when GPS is unavailable or as a cross-check
- Aviation flight planning and in-flight position estimation
- Submarine navigation using inertial navigation systems
- Robot localization in GPS-denied environments (indoors, underground)
- Autonomous vehicle position tracking between GPS updates
- Hiking and wilderness navigation with compass and pace counting