Bearing Calculator: Initial Bearing, Midpoint and Rhumb Line

How to use it

Type the start in From and the place to head for in To: coordinates or city names.
Read Initial bearing, the direction to set off in, and Bearing on arrival.
For a destination, type the Bearing from the start and the Distance from the start and read Destination point.
Compare Rhumb-line bearing and Rhumb-line distance if you steer a constant compass course, and open Working for each formula.

Initial and final bearing, midpoint and destination

On a great circle, the shortest path over the sphere, the bearing changes all along the route. The initial bearing is the direction at the start. The final bearing is the direction of travel at the end: the reverse of the bearing from the end back to the start. The formulas treat the Earth as a sphere of mean radius 6 371.0088 km, as Movable Type Scripts gives them.

θ = atan2(sin Δλ × cos φ₂, cos φ₁ × sin φ₂ − sin φ₁ × cos φ₂ × cos Δλ)
Bx = cos φ₂ × cos Δλ, By = cos φ₂ × sin Δλ
φm = atan2(sin φ₁ + sin φ₂, √((cos φ₁ + Bx)² + By²)), λm = λ₁ + atan2(By, cos φ₁ + Bx)
φ₂ = asin(sin φ₁ × cos δ + cos φ₁ × sin δ × cos θ), with δ = distance ÷ R

A bearing of 0 is north, 90 east, 180 south and 270 west.
The midpoint is the halfway point along the great circle. It is not the average of the two latitudes and longitudes.

Rhumb lines

A rhumb line, or loxodrome, crosses every meridian at the same angle, so it keeps one bearing all the way: it is the course a compass steered at a fixed heading follows. On a Mercator chart it is a straight line. It is longer than the great circle: for an east-west course at 20° north it is about 1.5 % longer, and at 60° north about 8.5 % longer (Wikipedia, Rhumb line).

Δψ = ln(tan(45° + φ₂/2) ÷ tan(45° + φ₁/2)), the isometric latitude difference
bearing = atan2(Δλ, Δψ)
distance = R × √(Δφ² + q²Δλ²), with q = Δφ ÷ Δψ (q = cos φ₁ on a parallel)

A rhumb line runs into a pole in a spiral, so no rhumb-line result is given near a pole.

What this tool does not cover

Magnetic bearings. The bearings here are true, from true north; magnetic variation is shown on charts and must be applied before you steer.
Winds, currents and heights. The results are for the surface of the sphere.
Ellipsoidal bearings. The distance tool gives the WGS 84 (Vincenty) distance and bearing.

Worked example: Istanbul to Ankara, and 300 km on 45°

From 41.0082, 28.9784 to 39.9334, 32.8597 the initial bearing is 108.73°, and the direction of travel on arrival is 111.25°. The halfway point is 40.487031, 30.934586.

Set off from Istanbul on a bearing of 45° and travel 300 km along the great circle: you reach 42.887141, 31.582071, still heading 46.74°.

Result	Value
Initial bearing	108.73°
Bearing on arrival	111.25°
Midpoint	40.487031, 30.934586
Rhumb-line bearing	110.00°
Rhumb-line distance	349.38 km
Destination (45°, 300 km)	42.887141, 31.582071
Bearing at the destination	46.74°

Frequently asked questions

Why does the bearing change along a great-circle route?

A great circle is the shortest path over the sphere, and it cuts the meridians at a changing angle as it goes round the Earth. A ship or aircraft that keeps one compass heading follows a rhumb line instead, which is longer but needs no change of course.

What is the difference between the initial and the final bearing?

The initial bearing is the direction to set off in, and the final bearing is the direction of travel when you arrive. On a long route the two can differ by many degrees: from Valparaíso to Shanghai the bearing changes from about 266° to about 282°.

Is a magnetic compass bearing the same as these bearings?

No. These bearings are true, measured from true north. A magnetic compass points to magnetic north, which differs from true north by the local magnetic variation shown on the chart. Apply that variation, east or west, before you steer.

Why is there no rhumb line near the North or South Pole?

Every meridian meets at a pole, and a rhumb line that approaches it winds round in a spiral and never quite arrives. The tool gives the great-circle values there, but no rhumb-line bearing or distance.

How accurate are these bearings?

They use a sphere of mean radius, which Movable Type Scripts puts at up to about 0.3 % error for distances. For a survey or a legal boundary, use the ellipsoidal bearing and distance of the distance tool, which follow the WGS 84 shape.

Can I type a city name instead of coordinates?

Yes. Names come from the GeoNames list of places with more than 15 000 people, and case and accents do not matter. Add the country code after a comma, such as Ankara, TR, when a name is shared by more than one place.

Sources

Figures and text last checked on Tuesday, November 10, 2026.

Result

Working