Waypoint Dead Reckoning Calculator

Dead reckoning is the process of estimating your current position by starting from a known location and applying your direction of travel and distance covered. It is one of the oldest navigation techniques, used for centuries before GPS by sailors, pilots, and explorers.

Distance = Speed × Time, in any consistent pair of units.

Turning that distance and a heading into a new latitude and longitude is where it gets interesting. Holding one compass heading traces a rhumb line, a spiral that cuts every meridian at the same angle, and the sums for it are these:

• Δlat = distance × cos(heading) / R (in radians, R = 6,371 km)
• New latitude = start lat + Δlat
• q = Δlat / Δψ, where Δψ is the difference in stretched latitude, ln(tan(45° + lat/2)), between the two points
• Δlon = distance × sin(heading) / (R × q)

That q is the whole trick, and it is what a naive version gets wrong. The obvious shortcut is to divide the easting by cos(start latitude), since a degree of longitude shrinks as you go poleward. But you do not stay at the start latitude. Over a leg heading northeast from London you finish more than a degree further north, where the meridians are already closer together, and using the departure cosine for the whole run overstates how far east you got.

The error is not academic. On a 500 km leg from 60°N on a heading of 060, the shortcut puts you 15 km off the true rhumb-line position. That is 3% of the leg, not the fraction of a percent people assume. Even a modest 185 km leg out of London on 045 lands about 1.8 km wrong. This calculator uses the stretched-latitude form, which is exact on a sphere, so the error is only the sphere-versus-ellipsoid difference of a few parts in a thousand.

Rhumb line, not great circle. A great circle is shorter, but you cannot steer it with one number, and dead reckoning is by definition “I held this heading for this long.” Over 500 km the two answers differ by about 25 km, so it matters which one you are computing. This page gives the constant-heading answer, and reports the great-circle position alongside it so you can see the gap.

Sources of error in dead reckoning:

1. Speed error: your actual speed differs from your estimate, thanks to current, wind, or terrain
2. Heading error: 1° off at 100 km puts you 1.75 km off course
3. Accumulated error: each leg starts from the last leg’s estimate, so the errors stack rather than average out
4. Leeway: wind and current push you sideways, so your track is not your heading

This is why navigators historically fixed their position at every opportunity, off landmarks, star sights, or radio beacons, and then restarted the dead reckoning from the new known point. The fix does not correct the past; it resets the future.

Modern use: despite GPS, dead reckoning has not gone away. Submarines run on inertial navigation, which is dead reckoning with very good accelerometers. Aircraft carry it as a backup. Hikers use pace counting on a bearing. Autonomous vehicles dead-reckon between GPS updates, and in a tunnel it is all they have.