How this instrument works
Azimuth, or bearing, is a direction measured clockwise from true north, running from 0° (due north) through 90° (due east), 180° (due south) and 270° (due west) back to 360°. Between two points on a flat map that would be a single fixed number, but the Earth is a sphere, and the shortest path between two points on a sphere — the great-circle route — generally curves in a way that makes the bearing change continuously as you travel along it. What this calculator returns is the initial bearing: the exact heading you'd set out on at the very start of the journey.
The underlying formula comes from spherical trigonometry, combining the sines and cosines of both latitudes with the difference in longitude, then resolving the result with an atan2 function so the bearing lands in the correct compass quadrant. It's the same relationship used in maritime and aviation navigation, where 'great-circle bearing' and 'initial bearing' are standard terms for exactly this quantity.
This matters because a straight line drawn on a common flat map (a Mercator projection) is almost never the shortest real-world path, and its apparent direction is almost never the correct compass heading for a long trip. A flight from New York to London, for instance, looks like it should head due east on a flat map, but the true shortest path — and the bearing this calculator returns — cuts noticeably north of that, tracking the curve of the sphere.
- Enter Start latitude and Start longitude in decimal degrees — negative values for south latitudes and west longitudes.
- Enter Destination latitude and Destination longitude the same way.
- Read Initial great-circle bearing — the compass heading, from 0° to 360°, to set out on from the start point.
- Remember the bearing changes continuously along a long great-circle route; recompute using a point partway along the journey if you need the heading partway through instead of at the very start.
Worked example — New York City to London
Enter 40.7128 and −74.006 as the start latitude and longitude (New York City), and 51.5074 and −0.1278 as the destination (London). Initial great-circle bearing reads 51.21 degrees — a heading well north of due east, which is why transatlantic flight paths departing the US East Coast curve noticeably northward rather than following a straight line drawn on a flat map.
Questions
Why isn't the bearing simply 'east' for a trip like New York to London?
Because the Earth is a sphere, not a flat map, and the shortest path between two points at similar latitudes still curves toward the nearer pole. A straight line on a Mercator map looks like due east, but the true shortest route — the great circle — bends northward through higher latitudes before coming back down, which is exactly why the New York-to-London bearing comes out around 51°, well north of 90° (due east).
Does the bearing stay the same for the whole journey?
No, except in two special cases: traveling along the equator, or traveling due north or south along a single meridian. On every other great-circle route the bearing changes continuously as you travel, which is why this calculator specifically returns the initial bearing — the heading at the very start of the trip, not a constant heading for the whole way.
What's the difference between a great-circle bearing and a rhumb-line bearing?
A great-circle bearing follows the shortest possible path between two points but requires continually adjusting your heading as you travel it. A rhumb line (or loxodrome) instead holds a single constant compass bearing the entire way, which is easier to steer by but covers a longer distance. Ocean and air navigation historically used rhumb lines for simplicity and great circles for efficiency on long-haul routes.
Why does going due east along the equator give exactly 90 degrees?
Along the equator, the great-circle path and the 'straight east' path are the same line, so there's no curvature to account for and the bearing stays fixed at exactly 90°, due east. This is a useful sanity check for the formula: starting and ending at latitude 0° with only longitude changing always returns a clean 90° or 270°, depending on direction.
Can I use this for real flight planning or navigation?
It gives the correct geometric starting heading for the great-circle route between two coordinates, which is the same relationship pilots and navigators use as a baseline. Actual flight plans also have to account for magnetic declination, prevailing winds, air traffic routing and airspace restrictions, so treat this calculator's output as the geometric starting point, not a substitute for a filed flight plan.