What Is a Rhumb Line?

Q: What is a rhumb line?

A rhumb line — also called a loxodrome — is a path on the Earth that crosses every meridian at the same angle. Following a constant compass bearing produces a rhumb line. On a Mercator projection map, rhumb lines appear as straight lines (the defining property of Mercator); on a globe, a non-trivial rhumb line spirals slowly toward the pole. Rhumb lines are usually longer than the corresponding great-circle paths, but they are easier to steer because the bearing doesn't change.

Q: Why is a rhumb line different from a great circle?

Because they're defined differently. A great circle is the shortest path on a sphere — the unique circle through both points whose plane passes through the centre. A rhumb line is the path of constant compass bearing — at every point, the bearing to the destination is the same. The two coincide only along meridians, along the equator, and over very short distances. For all other paths, the rhumb line is the longer of the two — sometimes negligibly, sometimes substantially (~5–10 % on long transoceanic paths).

Q: Why is the rhumb line a spiral?

Because meridians converge at the poles. To maintain a constant angle with each meridian, the path must turn faster as it approaches the pole. The result is a spiral that approaches the pole asymptotically — never actually reaching it. Mathematically, the loxodrome is a transcendental curve described by polar coordinates; the spiral makes infinitely many turns around the pole before converging. In practical navigation this matters only for paths bound for very high latitudes; for typical journeys the path looks roughly like a Mercator-map straight line.

Q: Are rhumb lines still used in modern navigation?

Yes, in three contexts. First, marine navigation in confined waters (channels, harbours, fjords) where a constant compass bearing is easier to steer than a continuously varying great-circle bearing. Second, short-distance legs (under ~500 km) where the rhumb / great-circle distance difference is negligible. Third, traditional chart-and-compass navigation that doesn't have access to GPS / FMS computed routes. For long-distance flight and shipping, modern computer-managed routing follows the great-circle path because the savings are real.

Q: What is the formula for rhumb-line distance?

The rhumb-line distance is computed from latitude differences and the constant bearing: d = R · sec(α) · |Δφ|, where α is the constant bearing (measured from north), Δφ is the latitude difference in radians, and R is the Earth radius. When the bearing is east or west (cos α = 0), the formula degenerates and a different expression involving the longitude difference and the latitude is used. The formula is closed-form (no iteration needed) and gives results within ~0.5 % of the true rhumb-line distance on the WGS 84 ellipsoid.

A rhumb line (loxodrome) is a path on Earth that crosses every meridian at the same angle — a constant compass bearing. On a globe it spirals toward the pole; on a Mercator map it appears as a straight line, which is the property Gerardus Mercator engineered his 1569 projection to produce. The article covers the definition, the geometry, the rhumb-vs-great-circle distance comparison, the closed-form distance formula, and when rhumb lines are still preferred (marine navigation, short legs).

In 1569, the maritime world had a problem: ships at sea could hold a compass bearing accurately, but the maps of the era distorted that bearing into curves that were impossible to plot mechanically. Gerardus Mercator's solution was a new map projection in which any constant-bearing path — a rhumb line — appears as a straight line on the chart. Plot the line, read the bearing once, steer that bearing all the way. The Mercator projection became (and remains) the standard maritime navigation projection precisely because of this property.

The great-circle-distance pillar covers the shortest path between two points. This article covers the rhumb line — the constant-bearing path that is the practical alternative.

Definition and geometry

A rhumb line (Portuguese rumbo, Spanish rumbo, from the Latin rhombus) is a path on Earth's surface along which the angle between the path and each meridian is constant. Equivalently, it's the path produced by steering a constant compass bearing relative to true north.

On a sphere:

A rhumb line on a meridian (bearing 0° or 180°) is a great circle: the meridian itself. The rhumb and great-circle paths coincide.
A rhumb line on the equator (bearing 90° or 270°) is also a great circle: the equator. Again the two coincide.
A rhumb line at any other bearing is a loxodrome — a spiral that crosses meridians at a constant angle and approaches one of the poles asymptotically.

The poles themselves are limit points: a rhumb line at bearing other than 0°, 90°, 180°, 270° loops around the pole infinitely many times as it approaches, without ever reaching it. In practice the spiral takes most of its length far from the pole, so the “infinite spiral” pathological behaviour only matters for routes intended to reach polar regions (in which case great-circle or polar-azimuthal routing is preferred anyway).

The mathematics

For a sphere of radius R, the rhumb-line distance from (φ₁, λ₁) to (φ₂, λ₂) is computed by integrating along the constant-bearing path:

α = arctan2(Δλ, ln(tan(π/4 + φ₂/2) / tan(π/4 + φ₁/2)))   (constant bearing)
d = R · √((Δφ)² + (cos(φ_mean) · Δλ)²)                    (approximate)

The exact integral form for any constant α:

d = R · (Δφ) / cos(α)

When α is near 90° (east or west bearing), cos(α) is near zero and the formula degenerates. The proper formula along an exact east-west bearing is d = R · cos(φ) · Δλ (the small-circle distance at constant latitude).

For navigators, the practical computation goes:

Compute the constant bearing α from the formula above.
Compute the distance using d = R · Δφ / cos(α) (when valid) or d = R · cos(φ) · Δλ (when the path is nearly east-west).

The Coordinately tools currently compute great-circle distance via Vincenty (see /tools/distance-calculator and /learn/vincenty-formula-explained); rhumb-line distance is left to specialised marine-navigation software.

Rhumb-line vs great-circle: distance difference

The rhumb-line path is always at least as long as the great- circle path, equal when both paths coincide (along meridians and the equator), longer otherwise. The percentage difference grows with:

The angular separation between the endpoints.
The latitude difference.
The bearing's deviation from 0° or 90°.

Some worked examples:

| Path | Great-circle distance | Rhumb-line distance | Difference | | ----------------------------- | --------------------- | ------------------- | ---------- | | NYC → London (5,585 km) | 5,585 km | 5,777 km | 192 km (3.4 %) | | LA → Tokyo (8,790 km) | 8,790 km | 8,820 km | 30 km (0.3 %) | | NYC → Hong Kong (12,990 km) | 12,990 km | 16,200 km | 3,210 km (24.7 %) | | London → Sydney (16,990 km) | 16,990 km | 21,750 km | 4,760 km (28.0 %) | | Empire State → Statue of Liberty (4,838 m) | 4,838 m | 4,838 m | < 1 m |

The NYC → Hong Kong and London → Sydney pairs show why the great-circle path matters for transoceanic flights: the rhumb- line alternatives are 25–28 % longer. For LA → Tokyo, the rhumb and great-circle paths are nearly equal because both cities are at similar latitudes (~35°N) and the path is mostly east-west.

Historical importance

Before Mercator's 1569 projection, navigators worked with portolan charts — flat maps with rhumb-line networks drawn across them as a starburst pattern from compass roses at key ports. Captains would identify two ports along a single rhumb line, set the bearing of that line, and steer it. The portolan system worked but was limited: charts had to be locally calibrated, and accurate inter-port distances depended on trial-and-error empirical correction.

Mercator's projection solved the problem mathematically: every constant-bearing path appeared as a straight line on his chart, and the chart was internally consistent at the global scale. A navigator could draw a single straight line between two distant ports, measure the bearing, set the compass, and sail — with no further adjustment until landfall (modulo currents and wind). The Mercator chart effectively turned global navigation into ruler-and-compass geometry.

For 350 years, marine navigation operated almost entirely on rhumb-line tracks plotted on Mercator charts. The transition to great-circle navigation only happened in the 20th century:

1859 — Inman's navigation tables include haversine for great-circle calculation, but the practical use case is limited to academic navigation training.
1920s–1940s — Trans-Pacific shipping begins systematic use of composite tracks (alternating great-circle and rhumb- line segments) to combine the shortest-path advantage with manageable compass steering.
1950s onward — Commercial aviation adopts great-circle routing as routine. Inertial navigation systems and (later) GPS make constant-bearing steering unnecessary.
2000s — Modern container shipping uses computer-optimised weather routing that approximates great-circle paths with jet-stream and current adjustments. Rhumb lines remain the default for short coastal navigation and in confined waters.

When rhumb lines are still preferred

Three contexts:

Marine navigation in confined waters. In channels, fjords, or harbour approaches, a rhumb line is easier to steer with a compass than a continuously-changing great-circle bearing. The practical question is whether the navigator can react fast enough to small bearing changes on a manual compass; for short distances and confined-channel work, the rhumb-line approach is simpler and safer.

Recreational sailing and aviation. Many sailors and pilots plot rhumb-line courses on their charts for simplicity. For short legs and casual cruising, the few-percent distance penalty is insignificant compared with the ease of steering. Bluewater sailors crossing oceans typically run waypoint-segmented routes that approximate the great-circle path with rhumb-line segments between waypoints.

When charts are paper-only. Aircraft carriers, military units operating under signal-denial scenarios, and traditional sailing vessels may operate without GPS/FMS computed great-circle routing. In these cases, rhumb-line plotting on a Mercator chart is the practical default.

For modern commercial shipping and aviation, computer-routing software computes the great-circle baseline and adjusts for winds, currents, weather, and operational constraints. The rhumb line is a fallback for cases where the computer is unavailable.

Mercator's straight rhumb line in practice

A worked example. On a Mercator-projected world map, draw a straight line from New York (40.71°N, 74.01°W) to Cape Town (33.92°S, 18.42°E). The line crosses the equator at roughly 21°W. Measure the bearing: about 119° (south-east). Follow that bearing in real life and you trace the rhumb line — a 12,700 km path that passes through the Atlantic Ocean roughly along the charted track.

The great-circle path between the same two cities is shorter: about 12,540 km, with an initial bearing of about 121° and a final bearing of about 154°. The great-circle path crosses the equator further west and passes closer to the Brazilian coast before turning toward the South African coast. The two paths diverge by up to a few hundred kilometres in the middle of the ocean before converging at the destination.

For a single ship on a single voyage, the 160 km rhumb-line penalty (~1.3 %) might or might not be worth the steering complexity. For a shipping line running the route weekly with heavy fuel costs, every percent of distance saved compounds — which is why modern shipping uses great-circle routing with weather routing layered on top.

Common misconceptions

“Rhumb lines are obsolete.” Far from it. Marine navigation still uses them constantly in coastal waters. The Mercator projection is still the dominant marine chart. The rhumb-line model is the default for short-leg and confined-water work.

“Rhumb lines always spiral around the poles.” The spiral behaviour applies to non-trivial bearings. A rhumb line along a meridian (north-south) is a great circle, not a spiral. A rhumb line along the equator (east-west) is also a great circle. Only intermediate bearings produce the loxodrome spiral.

“The Mercator straight line is the actual physical straight line.” It's straight on the Mercator chart; on the actual curved Earth, it's the loxodrome spiral. The straight-line appearance is the projection's feature, not a property of the physical path.

“Rhumb-line and great-circle paths give the same answer at short distance.” Approximately yes for distances under ~500 km — the two paths are indistinguishable to the eye and the distances agree to better than 0.1 %. The difference grows quadratically with distance.

“Modern GPS makes rhumb lines unnecessary.” GPS makes great-circle routing trivial, but rhumb lines remain useful in their specific niches. A small-boat sailor in a harbour still steers by compass; a backup navigator on an aircraft can still plot a rhumb-line track if the FMS fails. The rhumb-line model isn't obsolete; it's a specialised tool.

“Loxodromes are mathematical curiosities.” They were the practical foundation of pre-GPS navigation for centuries. The modern reduction of their importance is recent (last 60–70 years); the curve itself has a 450-year history of practical use.

“A rhumb line and a great circle are the same on a globe at short distances.” They're indistinguishable to the eye on a globe at any distance up to about 1,000 km, and mathematically agree to better than 0.1 % at that scale. The difference only becomes visually apparent at thousand-km scales, where the great-circle path's subtle northward curvature becomes visible against the rhumb line's straighter (in constant-bearing terms) track.

Great-Circle Distance— The pillar — the shortest-path alternative
The Mercator Projection— The projection engineered to make rhumb lines straight
Why Flight Paths Look Curved— The visual contrast between rhumb and great-circle on a map
Great-Circle vs Rhumb Line— Side-by-side comparison (when shipped)
Methodology— How content is sourced and verified

Frequently asked questions