---
title: "Bearing and Midpoint Between Coordinates | North Pinnacle"
description: "Initial and final great-circle bearing between two points, the compass point, the rhumb-line course and the geographic midpoint, drawn on a compass rose."
url: https://northpinnacle.top/bearing-and-midpoint/
---

Sheet 3 · Bearing and midpoint

# Bearing and midpoint between two points

Which way to set off, which way you'll be heading on arrival, the constant-compass alternative, and the point halfway along the route.

## Readout

initial final rhumb line

## Why the bearing changes on the way

The shortest route between two points follows a great circle, and on a great circle your heading keeps turning unless you travel along a meridian or the equator. The **initial bearing** is the direction to set off in, measured clockwise from true north:

θ = atan2(sin Δλ · cos φ₂, cos φ₁ · sin φ₂ − sin φ₁ · cos φ₂ · cos Δλ)

From London, Tokyo lies at about 032° (north-northeast, across Scandinavia and Siberia), yet you arrive heading about 156°. A **rhumb line** keeps one compass course, 099° here, and draws a straight line on a Mercator chart, but it is about 1,700 km longer.

The **midpoint** is halfway along the route, not the average of the latitudes and longitudes. London–Tokyo passes its halfway point near 69°N, inside the Arctic Circle.

## True or magnetic?

- These are true bearings, from the geographic North Pole.
- A hand compass points to magnetic north; add the local east declination to a magnetic bearing to get a true one.
- Look up today's declination with the [NOAA magnetic field calculator](https://www.ngdc.noaa.gov/geomag/calculators/magcalc.shtml).

## Sources

- Veness C. [Calculate distance, bearing and more between latitude/longitude points](https://www.movable-type.co.uk/scripts/latlong.html), Movable Type Scripts: the bearing, midpoint and rhumb-line formulas, and the Land's End to John o' Groats example used in this page's tests.
- Karney CFF. Algorithms for geodesics. *Journal of Geodesy* 87 (2013), 43–55. [doi:10.1007/s00190-012-0578-z](https://doi.org/10.1007/s00190-012-0578-z): the ellipsoidal azimuths and midpoint.

## Questions

**What is the difference between final and back bearing?**

The final bearing is your heading as you arrive. The back bearing is the initial bearing turned round by 180°. They match only on meridians and the equator.

**Can I follow these bearings with a hiking compass?**

Over a few kilometres, yes, after correcting for magnetic declination. On a long route the great-circle heading drifts, so take a new bearing from each waypoint.

**Why are there two midpoints?**

One is halfway along the geodesic on the WGS84 ellipsoid, the other halfway along a great circle on a sphere. They usually differ by a few kilometres on long routes.