Coordinates, great-circle distance, and bearings

Why do flights from London to Tokyo head northeast over Scandinavia instead of due east? Why is the halfway point between two cities not the average of their coordinates? The answers come from one fact: the Earth is round, and the shortest route between two points on a sphere is a curve on a flat map. This guide covers coordinate notation, great-circle distance and bearing, and the mistakes that put points in the wrong place.

Latitude, longitude, and notation

Latitude measures north or south of the equator, from −90° to +90°. Longitude measures east or west of the Greenwich meridian, from −180° to +180°. GPS, web maps, and most data use the WGS84 system and decimal degrees: Amsterdam is at 52.3676, 4.9041.

The same position in degrees, minutes, and seconds is 52° 22′ 3.36″ N, 4° 54′ 14.76″ E. Each degree has 60 minutes and each minute 60 seconds. The coordinate converter switches between the notations. Useful precision rules:

• One degree of latitude is about 111 km, and one minute of latitude is one nautical mile (1,852 m).
• Five decimal places of a degree is about 1.1 m, which is enough for most uses. Six is about 11 cm.
• A degree of longitude shrinks with latitude: 111 km at the equator, about 68 km at Amsterdam’s latitude, and zero at the poles.

Great-circle distance

A great circle is any circle around the globe whose centre is the centre of the Earth, like the equator or a meridian. The shortest path between two points on a sphere always follows one. The distance calculator uses the haversine formula with the mean Earth radius of 6,371.0088 km:

• Amsterdam to Paris: 429.9 km (267.1 miles, 232.1 nautical miles).
• London Heathrow to Tokyo Haneda: 9,591.6 km.

These are straight-line distances over the surface, “as the crow flies”. Road and rail routes are typically 20–40% longer, and flights deviate for winds, airspace, and airways. Treating the Earth as a perfect sphere introduces an error of up to about 0.5% compared with the more precise ellipsoidal model. That is negligible for planning and comparisons, but surveying and legal boundaries need ellipsoidal geodesic calculations.

Bearings change along the way

The bearing calculator gives the initial heading, measured clockwise from true north. From London Heathrow to Tokyo it is 31.4°, north-northeast, even though Tokyo is at a lower latitude than London. The great circle bends north toward the pole and curves back down, so the heading changes continuously along the route. Only paths due north, due south, or along the equator keep a constant heading.

A route with a constant compass heading is called a rhumb line. It is easier to steer but longer, which is why ships and aircraft on long routes follow great circles in segments. Over short distances the difference disappears: from Amsterdam to Paris the initial bearing is 205.8° and the final bearing 203.8°.

Bearings here are true bearings. A magnetic compass points to magnetic north, which differs by the local declination, several degrees in many places and changing each year. Correct for it before navigating with a compass.

Midpoints and destination points

The midpoint calculator finds the point halfway along the great circle. Between London and New York it lies at 52.4° N, 41.3° W, further north than either city, while the simple average of the coordinates is about 760 km away from it. For nearby places, such as Amsterdam and Paris (midpoint 50.62° N, 3.58° E), the two methods almost agree.

The destination point calculator works the other way: from a start, an initial bearing, and a distance, it finds where you end up. It is useful for plotting a search area, placing test points at known distances, or checking a navigation log. Small heading errors grow with distance. Over 5,570 km, a third of a degree moves the endpoint by about 28 km.

The coordinate-order trap

People and many APIs say “latitude, longitude”. GeoJSON, many GIS tools, and WKT use “longitude, latitude”, which corresponds to x, y. Swapping them is the single most common geodata bug. It usually puts a European point in the Indian Ocean or off the coast of Somalia. When a point lands somewhere absurd, check the order first. A latitude outside ±90 is a certain sign of a swap.

Before loading GeoJSON into a map, run it through the GeoJSON validator, which checks structure, coordinate ranges, and polygon closure. Use GeoJSON bounds to get the bounding box that map libraries use to fit the view. GPS tracks from devices and fitness apps can be turned into spreadsheet rows with the GPX waypoint extractor.