Great circle route map

Worked example: New York to Tokyo

Using 40.7128, -74.0060 for New York and 35.6762, 139.6503 for Tokyo, the great circle leaves New York on a bearing of 333.0° (NNW) and reaches Tokyo on 205.1°. On the way it climbs to 69.9° north, far above both cities, which is why the route arcs far to the north on a flat map.

The rhumb line holds a single heading of 267.5° and stays at low latitudes. It is 1,206 miles longer.

Why the shortest route looks curved

Mercator maps stretch the map east to west more and more as you move away from the equator. A straight line on Mercator is a line of constant compass heading, called a rhumb line or loxodrome. It is easy to steer but it is not the shortest path.

The shortest path lies on a great circle, the intersection of the globe with a plane through the Earth's center. On the orthographic globe it appears as a smooth arc. Drawn on Mercator it bows toward the nearer pole, which is why flight maps show curved routes.

Reading the numbers

The readout gives both lengths and their difference in miles and as a percentage. For short trips or trips that run mostly north to south, the two routes are nearly identical. The gap grows with distance and with latitude, and is largest for long east-west trips far from the equator.

The northernmost point is the highest latitude the great circle segment reaches. When that is inside the trip, it is the vertex of the great circle, found from Clairaut's relation: the cosine of the vertex latitude equals the sine of the starting bearing times the cosine of the starting latitude. When the vertex lies beyond one of the endpoints, the readout says so and reports the endpoint.

Limits of the model

Both routes are computed on a sphere of radius 6,371.0088 km. Airlines plan on the WGS84 ellipsoid and then bend routes around winds, airspace, and airports, so a real flight track is longer than the great circle shown here. Use this as the geometric baseline, not a flight plan. For routes of a few hundred miles the two lines nearly coincide, so the comparison is most telling for long trips across oceans or continents.

Two more routes, computed

Between the farthest pair of state capitals, Honolulu, HI and Augusta, ME, the great circle is 5,111 miles and the rhumb line 5,295 miles, a difference of 3.6%. The great circle reaches 46.1° north on the way.

A mostly north-south trip shows the opposite case. From Juneau to Phoenix the great circle is 2,005 miles and the rhumb line 2,012, only 0.34% longer, because a path close to a meridian is already nearly a great circle.