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.

Input

Readout

initialfinalrhumb line

Initial bearing: 031.7°NNE, north-northeast · ellipsoid 031.69°
Final bearing: 156.2°arriving at the destination · ellipsoid 156.24°
Back bearing: 211.7°the reverse of the initial bearing
Rhumb-line bearing: 099.0°constant course, 11,293.6 km (1,711.4 km longer than the great circle)

Midpoint

Halfway along the geodesic: 69.04348°N 89.58940°E69°02′36.5″N 089°35′21.8″E
Great-circle midpoint (sphere): 68.99099°N 89.63597°E68°59′27.5″N 089°38′09.5″E

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.

Sources

Veness C. Calculate distance, bearing and more between latitude/longitude points, 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: 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.