Great Circle Calculator

What Is a Great Circle and Why Is It the Shortest Path

A great circle is the largest possible circle that can be drawn on the surface of a sphere. It is formed by the intersection of the sphere with a plane that passes through the sphere's centre. On Earth, examples include the equator and any line of longitude (meridian). Every great circle divides the sphere into two equal hemispheres, and the arc connecting any two points along a great circle is the shortest possible surface path between them. This shortest surface path is called a geodesic, and on a sphere, every geodesic is a great circle arc.

In practical navigation, great circle routes matter most for east-west travel at high latitudes. A flight from New York to London, for example, does not fly in a straight line across the Atlantic at the same latitude; it arcs up over the North Atlantic toward Greenland. According to the SKYbrary aviation safety database, this polar arc saves 3 to 6 percent of the route distance compared to a constant-heading Mercator path, which for a 5,500 km transatlantic flight can amount to 200 to 330 km and several hundred kilograms of fuel per flight.

Great Circle vs Rhumb Line: Which Should You Use

A rhumb line, also called a loxodrome, is a path that crosses all meridians at the same angle. On a Mercator map projection, a rhumb line appears as a straight line between two points, which made it the preferred navigation method before GPS because sailors could plot a constant compass heading and maintain it throughout a voyage. The disadvantage is that a rhumb line is always longer than the corresponding great circle for east-west routes, sometimes significantly so.

The choice between methods depends on distance and direction. For routes running north-south, both methods coincide because meridians are both great circles and constant-heading paths. For east-west routes at low latitudes, the difference is small and the rhumb line is often practical. For long east-west routes at latitudes above 40 degrees, the great circle can save 5 to 10 percent of the journey distance. Maritime navigators often use composite routes: great circle to a point near the intended latitude, then a rhumb line along that latitude to avoid high-latitude ice and weather hazards. The International Maritime Organization guidance on ocean routing addresses exactly this trade-off between geodesic efficiency and operationally safe latitude limits.

How the Haversine Formula Works: Step by Step

The haversine formula computes great circle distance in four steps. First, convert all latitudes and longitudes from degrees to radians. Second, compute the haversine of the differences: a = sin²(Δφ/2) + cos(φ₁) × cos(φ₂) × sin²(Δλ/2), where φ is latitude and λ is longitude. Third, apply the inverse sine: c = 2 × arcsin(√a). Fourth, multiply by Earth's mean radius: d = R × c = 6,371.0088 km × c.

The haversine form is numerically preferred over the simpler spherical law of cosines (d = arccos(sin(φ₁)sin(φ₂) + cos(φ₁)cos(φ₂)cos(Δλ)) × R) because the cosine formula loses precision when the two points are very close together, where the cosine of a tiny angle is nearly 1 and small errors dominate the result. The haversine avoids this by working with sin²(half-angle), which stays well away from zero. According to Snyder's Map Projections: A Working Manual, the spherical approximation introduces an error of at most 0.3 percent compared to a true WGS84 ellipsoid calculation using Vincenty's formula, which is acceptable for navigation but not for geodetic survey work.

Initial and Final Bearing: Why They Differ on Long Routes

On a great circle route, the compass bearing changes continuously throughout the journey. The initial bearing is the direction you face at the start; the final bearing is the direction you are travelling when you arrive. These two values can differ by 30 to 60 degrees on long transatlantic or transpacific routes. For example, a flight from London to Los Angeles departs on approximately 320 degrees (NNW) and arrives on approximately 112 degrees (ESE), a total bearing change of over 150 degrees across the 8,750 km route.

This bearing change is not an error: it is the geometric consequence of following the geodesic on a spherical surface. In aviation, aircraft manage this by flying waypoints -- a series of intermediate coordinates where the heading is updated to follow the great circle. The intermediate waypoints table in this calculator shows exactly where each heading correction occurs and how the bearing evolves along the route. The FAA Aeronautical Information Manual describes how North Atlantic Track (NAT) waypoints are used to define composite great circle tracks for transatlantic traffic, with track codes updated daily to account for wind and traffic.

Intermediate Waypoints and Great Circle Navigation in Practice

Generating intermediate waypoints along a great circle uses spherical linear interpolation (SLERP). For a fraction t between 0 and 1, the waypoint coordinates are derived from: x = a·cos(φ₁)cos(λ₁) + b·cos(φ₂)cos(λ₂), y = a·cos(φ₁)sin(λ₁) + b·cos(φ₂)sin(λ₂), z = a·sin(φ₁) + b·sin(φ₂), where a = sin((1-t)σ)/sin(σ) and b = sin(tσ)/sin(σ), and σ is the central angle between the two points. These 3D Cartesian coordinates are then converted back to latitude and longitude.

In practical aviation and maritime navigation, waypoints serve multiple purposes: they allow air traffic control to track and separate aircraft, they enable the crew to cross-check their position against expected coordinates, and they provide handoff points between oceanic sectors. The ICAO standard for oceanic navigation requires at least hourly position reports, which correspond to roughly 700 to 900 km intervals on most transatlantic routes -- consistent with the 10-waypoint spacing this calculator generates by default.

Accuracy: Haversine vs Vincenty Formula

The haversine formula treats Earth as a perfect sphere and gives distances accurate to within 0.3 percent. For a 10,000 km flight, that is an error of at most 30 km. For navigation planning, route optimisation, and most engineering applications, this is entirely adequate. The Vincenty formula, developed by Thaddeus Vincenty in 1975, models Earth as a WGS84 oblate spheroid (flattening 1/298.257) and gives distances accurate to within 0.5 millimetres. It requires an iterative calculation that may not converge for nearly-antipodal points.

For surveying, geodesy, and legal boundary work, the Vincenty or Karney formulas are required. For aviation route planning, pilot training, search-and-rescue coordination, and distance estimation, haversine is the standard. The NOAA National Geodetic Survey inverse tool uses the Vincenty formula for professional geodetic work. This calculator uses haversine with Earth's mean radius of 6,371.0088 km (the IUGG recommended value), which gives a maximum error of approximately 22 km on antipodal routes and under 1 km for typical intercontinental distances.