Great circle vs rhumb line

Two ways to draw a straight line on a globe

On a flat sheet there is one straight line between two points. On a sphere there are two useful candidates. The great circle is the intersection of the globe with a plane through its center, the way a string pulled tight across a desk globe lies. The rhumb line, also called a loxodrome, is the path of constant bearing: a ship that never touches its wheel traces one.

The two coincide in only two cases: along a meridian, where both run due north or south, and along the equator. Everywhere else the great circle bows toward the nearer pole and the rhumb line runs closer to the equator.

How much longer the rhumb line is between capitals

The table lists the six longest pairs of capitals in the contiguous states plus Alaska, then five familiar routes. The extra distance is small on short or north-south trips and grows on long east-west trips at high latitude.

The formulas

The great circle distance is the haversine result explained in how distance is calculated. The rhumb line uses the stretched latitude of the Mercator projection, Δψ:

Δψ = ln(tan(π/4 + φ2/2) / tan(π/4 + φ1/2))
q  = Δφ / Δψ        (or cos φ1 when Δφ is near 0)
d  = R · √(Δφ² + q² · Δλ²)
bearing = atan2(Δλ, Δψ)

The Δψ term is why a rhumb line is a straight line on a Mercator map. Mercator stretches latitude by exactly that amount, which keeps every compass heading true on the chart. The Mercator distortion guide shows what the stretch does to areas.

Why flight paths look curved on maps

Most web maps use Web Mercator, where straight lines are rhumb lines. A great circle drawn on the same map arcs toward the pole, so a flight from the West Coast to Europe looks like a detour over Canada and Greenland. On a globe that arc is the straight path. The great circle route map draws both lines so you can see the difference for any pair.

Which one to use

• Straight-line distance between places: the great circle. This is what every distance tool on this site reports.
• Steering by hand with a compass over a short leg: the rhumb line, since the extra distance is tiny and the heading never changes.
• Long ocean or air routes: a great circle broken into several rhumb line legs, a practice navigators call great circle sailing.

How far north the great circle goes

A great circle between two northern points climbs to a highest latitude, its vertex, somewhere between them when the trip runs mostly east to west. From Sacramento to Boston the path peaks at 43.66° N, while the two ends sit at 38.58° and 42.36°. The rhumb line never goes north of the northern end, which is why it looks shorter on a flat map and is longer on the ground.

The same shape explains transatlantic flight tracks over Canada and Greenland and the polar routes between North America and Asia. The farther apart and the farther from the equator the two ends are, the higher the vertex climbs.

On short trips the difference vanishes

Between Boston and Providence the great circle and rhumb line differ by 4.6 centimeters. At that scale either line works, and a flat map is a fine guide. The choice only matters for trips of several hundred miles or more, and most of all for long east-west trips far from the equator.

Compare the two lines for your own trip

Enter two places in the great circle route map below. It draws the great circle and the rhumb line together, reports both distances, and gives the initial and final bearings of the great circle next to the single bearing of the rhumb line. If the two lines nearly overlap on the map, the trip is short or runs close to north and south, and the choice between them does not matter.