ABC Tools

Great Circle vs Rhumb Line

After reading this you will know why a flight from New York to Tokyo bends toward Alaska on the map, how to compute both the shortest path and the constant-bearing path between two points, and how to read the distances and bearings the tool reports.

Two honest ways to connect two points

Draw a line between New York and Tokyo on a flat map and it looks straight, running roughly west. Watch the actual flight path and it arcs high toward the north, brushing the Alaskan coast. Neither the map nor the airline is lying. They are answering two different questions.

The great circle is the genuinely shortest path over the surface of the sphere. A great circle is any circle whose center is the center of the Earth: the equator is one, every meridian is one. The short arc of the great circle through two points is the least distance between them, full stop.

The rhumb line, also called a loxodrome, is the path you get by holding one fixed compass bearing the whole way. It crosses every meridian at the same angle. On a Mercator map it draws as a perfectly straight line, which is exactly why Mercator was invented: a ruler on the chart gives you a course to steer.

So the curved arc on a normal map is the shortest route, and the straight line is the lazy-to-steer route. The curve is the map's distortion, not a detour.

When each one is the right answer

Use the great circle when distance or fuel is what you care about: flight planning, undersea cables, ballistic ranges, or just answering "how far apart are these two cities." Use the rhumb line when a constant heading matters more than a few extra kilometers: a small boat with a magnetic compass and no autopilot, or a quick sanity check of a bearing.

The two paths are nearly identical over short distances. Between two points 100 km apart the difference is a few meters. The gap grows with distance and with how far the route reaches from the equator. At the extreme, a rhumb line between two points can spiral endlessly toward a pole while the great circle takes a short hop straight across the top.

A rhumb line at constant bearing (other than due east, west, north or south) is a spiral that winds infinitely toward a pole. It has finite length but never quite arrives. That is a real property of the loxodrome, not a rounding glitch.

The formulas, and why they look the way they do

Both start from latitude and longitude in radians. Call the two points (\phi_1, \lambda_1) and (\phi_2, \lambda_2), with \phi for latitude and \lambda for longitude, and let R be the Earth radius.

Great circle distance

d = 2R \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta\phi}{2}\right) + \cos\phi_1 \cos\phi_2 \sin^2\left(\frac{\Delta\lambda}{2}\right)}\right)

This is the haversine formula. Here \Delta\phi = \phi_2 - \phi_1 and \Delta\lambda = \lambda_2 - \lambda_1. The term inside the square root is the "haversine" of the central angle. Taking 2R\arcsin(\sqrt{\cdot}) turns that angle into an arc length on the sphere. The haversine form stays numerically stable for very short distances, where a naive cosine formula loses precision.

Initial bearing of the great circle

\theta = \operatorname{atan2}\left(\sin\Delta\lambda \cos\phi_2,\ \cos\phi_1 \sin\phi_2 - \sin\phi_1 \cos\phi_2 \cos\Delta\lambda\right)

This gives the compass direction to set off in, measured clockwise from true north, then converted to the 0 to 360 degree range. The key fact: this bearing changes as you travel. That is what makes the great circle awkward to steer by hand and easy for an autopilot that recomputes constantly.

Rhumb line distance

d = R\sqrt{\Delta\phi^2 + q^2 \Delta\lambda^2}, \quad q = \frac{\Delta\phi}{\Delta\psi}

Here \Delta\psi = \ln\!\left(\tan\left(\tfrac{\pi}{4}+\tfrac{\phi_2}{2}\right) / \tan\left(\tfrac{\pi}{4}+\tfrac{\phi_1}{2}\right)\right) is the difference in "stretched latitude," the same stretch Mercator applies vertically. That logarithm is why a straight line on Mercator equals a constant bearing on Earth. When \Delta\psi is near zero (an east-west track), you replace q with \cos\phi_1 to avoid dividing by zero.

As two points move to higher latitudes while keeping the same longitude gap, the great circle bows more sharply toward the pole and the distance saving over the rhumb line grows. At the equator both paths coincide; near 60 degrees north the saving on a wide east-west span becomes hundreds of kilometers.

Worked example: New York to Tokyo

Reproducing the demo

The demo uses JFK at 40.6413° N, 73.7781° W and Haneda (HND) at 35.5533° N, 139.7811° E, with R = 6371 km.

  1. Convert to radians: \phi_1 = 0.7093, \lambda_1 = -1.2877, \phi_2 = 0.6205, \lambda_2 = 2.4397.
  2. Differences: \Delta\phi = -0.0888, \Delta\lambda = 3.7274. Note the longitude gap is more than half the globe, so it is reduced to the shorter side, giving an effective \Delta\lambda near -2.556 radians (about 146 degrees the other way).
  3. Haversine term: \sin^2(\Delta\phi/2) + \cos\phi_1\cos\phi_2\sin^2(\Delta\lambda/2) \approx 0.4076.
  4. Great circle distance: 2 \cdot 6371 \cdot \arcsin(\sqrt{0.4076}) \approx 10{,}870 km.
  5. Rhumb line distance from the loxodrome formula: \approx 12{,}000 km.
  6. Saving: about 1130 km, near 9.4 percent shorter.

The initial great circle bearing works out to roughly 333° (north-northwest), and the final bearing as you arrive at Tokyo is about 239° (west-southwest). That swing of nearly 94 degrees is the whole story: you leave heading well north of west and land heading south of west, because the route arced over the top.

Your exact numbers will shift by a few km if the tool uses a slightly different airport coordinate or a different Earth radius. The saving of just over 1,000 km on this route is robust to those choices.

Reading the results the tool gives you

The tool reports six things. Here is what each one means and how to sanity-check it.

Great circle length
The shortest surface distance. Divide by 6371 to get the central angle in radians if you want to reason about it as an angle.
Rhumb line length
Always greater than or equal to the great circle length. If they are equal, your two points share a meridian or both sit on the equator.
Distance saved
Rhumb length minus great circle length. Zero near the equator, largest for long high-latitude east-west runs.
Initial and final bearing
The great circle's compass heading at the start and at the end. On a rhumb line these two would be identical by definition; on a great circle they differ, sometimes a lot.
Northern- or southern-most point
The vertex of the great circle, where it briefly runs due east-west. Its latitude tells you how far poleward the route reaches. For a route in the northern hemisphere between two mid-latitude cities, the vertex sits higher than either endpoint.
The great circle is about 1,130 km shorter, roughly 9 percent, over this pair.

Common mistakes

Do not measure distance with a ruler on a Mercator map. Mercator stretches east-west spacing by 1/\cos\phi, so at 60 degrees latitude everything is twice as wide as at the equator. A straight ruler there measures a rhumb line, not the shortest path, and reads too long.

Three more traps trip people up. First, forgetting to reduce the longitude difference to the range from -180 to 180 degrees. If you leave \Delta\lambda at 214 degrees instead of using the shorter -146 degrees, you compute the long way around.

Second, mixing degrees and radians. Every trig function in the formulas expects radians. Feeding it 40 instead of 0.698 gives nonsense that is not always obviously wrong.

Third, treating the initial bearing as the whole-trip heading. It is not. On the New York to Tokyo route the bearing rotates by about 94 degrees end to end. Only a rhumb line keeps a single fixed heading.

Related tools

Once you have coordinates, a few other tools on this site pair naturally with route work. If your points came from a file, drop it into the GeoJSON & CSV Map Viewer to see them first. To switch a coordinate between decimal degrees, DMS, UTM or a geohash before you paste it, use the Coordinate Converter.

The Antipode Finder shows the point exactly opposite any location; a great circle through a point and its antipode can go any direction, since every such path is the same length. And for a decorative render of a route, the Old Map Renderer turns coastlines and lines into a parchment chart with a compass rose.

Frequently asked questions

Why do flights near the equator look straight?

Because near the equator the great circle and the rhumb line almost coincide, and Mercator barely distorts there. For a route from Singapore to Nairobi, both roughly along the equator, the two paths differ by well under 1 percent, so the shortest path looks nearly straight even on a flat map.

Do airlines actually fly the great circle?

Close to it, but not exactly. Real routes bend for jet streams, restricted airspace and air-traffic corridors. Flying with a strong tailwind can beat the geometric shortest path on time even while adding distance. The great circle is the floor on distance, not a promise of the exact track.

What is the vertex or "northernmost point" and why does it matter?

The vertex is where the great circle momentarily runs due east-west, its highest latitude. For New York to Tokyo the vertex sits well north of both cities, up near southern Alaska, which is why the arc appears to detour north. It marks the closest approach to the pole.

Which distance should I trust for "how far apart" two cities are?

The great circle distance. That is the standard "as the crow flies" figure and the number nearly every atlas and API reports. The rhumb line is a navigational convenience, always at least as long.

Why does a constant bearing not give the shortest route?

Because holding a fixed angle to every meridian forces the path to curve on the sphere. Meridians converge toward the poles, so keeping the same crossing angle means constantly turning relative to the true shortest direction. The saving is the price the sphere charges for that convenience.

Try it yourself in the tool

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