The Shortest Path on a Sphere
If you stretch a string taut between two points on a globe, it follows a great circle — the intersection of the globe with a plane passing through both points and Earth's center. This great circle arc is the shortest possible route between those two points on the surface. When airlines fly from New York to London, they arc northward over Newfoundland and Ireland rather than flying straight east on the map, because the great circle path is about 250 km shorter than the seemingly straight Mercator route.
Great Circles vs. Rhumb Lines
A rhumb line (loxodrome) crosses every meridian at the same angle, making it a straight line on a Mercator map and easy to navigate with a compass. But a rhumb line is almost never the shortest path — it spirals slightly, adding distance. The difference is negligible for short trips but can exceed 600 km for transpacific flights. Before GPS, sailors often navigated rhumb lines for simplicity and then corrected with great circle waypoints for efficiency.
The Haversine Formula
The haversine formula, published in 1805, calculates the great circle distance between two points given their latitude and longitude. It is numerically stable even for small distances (unlike the simpler spherical law of cosines) and remains the standard formula for distance calculations in navigation, aviation, and geospatial software. For precision on the WGS84 ellipsoid, Vincenty's formulae provide millimeter-accurate results.
Modern Flight Planning
Today's flight management systems compute great circle routes automatically and break them into a series of waypoints. Pilots fly each segment as a short rhumb line, approximating the great circle to within a few kilometers. Jet stream winds, restricted airspace, and ETOPS rules (for twin-engine overwater flights) cause deviations from the pure great circle, but it remains the starting point for every route calculation.