Primary sources · 4
- [1] Pedro Nunes (1537) — First to describe the rhumb line / loxodrome and prove it spirals to the poles rather than ending · Tratado da Sphera · 1537 https://en.wikipedia.org/wiki/Pedro_Nunes
- [2] Mercator (1569) — Original Mercator projection world map, designed to make rhumb-line courses appear straight · Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendate Accommodata · 1569 https://en.wikipedia.org/wiki/Mercator_1569_world_map
- [3] Bowditch — American Practical Navigator — Authoritative reference on rhumb-line and great-circle navigation · NGA Pub. 9, current edition · 2019 https://msi.nga.mil/Publications/APN
- [4] Veness — Movable Type Scripts — Working rhumb-line equations and JavaScript implementation · movable-type.co.uk/scripts/latlong.html · Continuously maintained https://www.movable-type.co.uk/scripts/latlong.html
A rhumb line is the course you would steer if you held one compass bearing for the whole trip. It spirals slowly toward whichever pole the bearing tilts to, and on a Mercator map it looks like a perfectly straight line — which is exactly why Mercator invented his projection.
The two ways to draw a line on Earth
There are two natural lines between two points on a sphere. The great circle is the shortest path — what a geodesic computes. The rhumb line (or loxodrome, from Greek "oblique course") is the path of constant true bearing — what a compass-steered ship draws. On a flat projection these two paths look very different; on a globe both are fully visible curves.
Why Mercator made rhumbs straight
Gerardus Mercator's 1569 projection stretches the latitude axis non- linearly so that every angle between any two intersecting curves on Earth's surface is preserved on the map. A constant-bearing course is by definition an angle-preserving curve relative to the meridians, and so shows as a straight line. The price is that area is wildly exaggerated near the poles — Greenland looks larger than Africa, when it is in reality 14 times smaller.
The excess varies with latitude
For two points at the same latitude on the equator, the great circle is the equator itself and the rhumb line is the same path — zero excess. For points at higher latitudes, the great circle bulges poleward while the rhumb line traces a curve closer to the parallel of latitude. The excess grows quickly with latitude and with the longitude span.
| Route | Great-circle | Rhumb | Excess |
|---|---|---|---|
| Equator: STI (Cape Verde) → JIB (Djibouti) | 6,760 km | 6,772 km | 0.2 % |
| JFK → LHR | 5,555 km | 5,597 km | 0.8 % |
| LAX → SYD | 12,051 km | 12,238 km | 1.6 % |
| JFK → HKG | 12,983 km | 15,231 km | 17.3 % |
| LHR → AKL | 18,330 km | 23,000 km | 25.5 % |
Why ships still use rhumb lines
Modern bulk carriers and most commercial maritime traffic plan rhumb-line courses for legs under about 1,000 nautical miles because steering one bearing is operationally simpler than re-plotting a great-circle multi-segment course every few hours. The extra distance is small on short legs and the navigation simplification is worth it. Beyond 1,000 nm the great-circle saving becomes large enough that ships break the course into shorter rhumb-line segments approximating the great circle.