Reckoning / Sky & sea / Great circle and rhumb line
Sky & sea · sailings
Great circle and rhumb line
There are two natural ways to cross an ocean. A rhumb line keeps the same compass course all the way: easy to steer, but longer. A great circle is the shortest path on the sphere, but its course changes continuously. Pick two ports and compare them.
Voyage
great circle rhumb line
The formulas
On a sphere of radius R, with latitudes φ, longitudes λ, the great-circle distance is computed with the haversine; the initial course from the spherical triangle formed with the pole. The rhumb line uses the meridional parts: the stretched latitude scale of the Mercator chart.
Distances use the navigator’s sphere, on which a minute of arc is exactly one nautical mile. The true distance on the WGS 84 ellipsoid comes from Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations, Survey Review 23 (176), 1975, the method checked here against its published worked example to the millimetre. The build also computes every great-circle distance twice, by the haversine and by vectors, and compares the rhumb-line formula with a step-by-step integration along the track.
Without computers, navigators drew the great circle as a straight line on a gnomonic chart, read off points every few degrees of longitude, transferred them to the Mercator chart and steered rhumb lines between them. The older mid-latitude sailing shortcut is shown in the table: good over short distances, poor over long ones. Today’s chart plotters do the same calculation on the ellipsoid.