How it works
A ship that holds one compass course sails a rhumb line (a loxodrome): it crosses every
meridian at the same angle. The shortest way between two places is a great circle (an
orthodrome; on the ellipsoid, a geodesic), and on it the course changes all the way. Each is
straight on some map and curved on the others, so the four maps here show the same two lines:
- Mercator is the map made so that rhumb lines are straight. That is why charts use it:
a ruler between two places gives the course to steer.
- Gnomonic projects from the centre of the earth, so every great circle (a plane through
the centre) is straight. Navigators drew the great circle on a gnomonic chart and moved its
points over to Mercator.
- Azimuthal equidistant on the start keeps distance and direction from the start true,
so the great circle is the straight line from the centre; this is the map a ship-centred view
of what is near uses. South polar stereographic (as EPSG:3031 is) is conformal, like
Mercator, so the rhumb line still crosses each meridian at its one angle; meridians there all
run into the pole, so the rhumb line becomes a spiral round it (on the sphere, a logarithmic
spiral). The whole loxodrome carries the course on, pole to pole: it winds round each
pole without end and is still finite in length.
The difference grows with distance and with latitude, which is why it is a daily matter in the
Southern Ocean. East or west at high latitude, the great circle bends towards the pole, and from
Hobart that means towards the ice. Keep north of a latitude is composite great circle
sailing, the navigator's answer: a great circle that just touches the limiting parallel, along it,
and a great circle on to the end. It is the shortest way that keeps north of the limit, and it is
usually much closer to the great circle than the rhumb line is.
Lengths and courses are on the WGS84 ellipsoid: the geodesic by Vincenty's method, the rhumb
line from the meridian distance and the isometric latitude (Snyder 1987), both checked against
PROJ in test/check-rhumb.mjs (the rhumb line there is PROJ's Mercator inverse of a
straight line, measured with PROJ's geodesics). The lines are drawn, and the composite sailing
worked, on the sphere of mean radius, which no map at these scales can tell apart.
Hash keys: alon, alat, blon, blat (the ends),
four (aeqd, polar), lox, comp,
lim.
- Snyder, J. P. (1987) Map Projections: A Working Manual, USGS Professional Paper 1395
(isometric latitude, meridian distance). Vincenty, T. (1975) Direct and inverse solutions of
geodesics on the ellipsoid, Survey Review 23, 88. Karney, C. F. F. (2013) Algorithms for geodesics,
J. Geodesy 87, 43, as PROJ and
GeographicLib have them.
- The ship-to-target crow lines of the nuyina-proximal
tracker, and the Nuyina page, which draws both lines from the ship.
- Maps drawn with d3-geo; coastlines from
Natural Earth.