Rhumb Line Course and Distance Calculator

Two ways to cross an ocean

When planning a voyage across the open ocean, navigators have two fundamental choices:

Great Circle (Orthodrome): the shortest path between two points on Earth’s surface. Like a straight line on a globe.

Rhumb Line (Loxodrome): a path of constant compass bearing. Like a straight line on a Mercator chart.

These are different paths. The great circle is shorter but curves on a flat chart. The rhumb line is longer but appears as a straight line on Mercator projections.

On a short coastal hop the two are the same to within a rounding error. On a long crossing the gap can be anything from nothing at all to a third of the distance, depending on which way the route runs. The table further down shows six real routes and how far apart they land.

The rhumb line explained

A rhumb line (also called loxodrome from Greek “slanted course”) crosses every meridian at the same angle. On the surface of the Earth, this produces a spiral path toward the poles. Locally, over any short segment, it’s a straight line on a Mercator chart.

The constant-bearing property is what makes rhumb lines navigationally useful:

• Set compass to the rhumb course
• Steer that exact bearing throughout the voyage
• Never need course corrections
• Easy for human navigators with no GPS

This simplicity is why pre-GPS sailors used rhumb lines for ocean crossings despite knowing they were longer.

The math behind rhumb lines

The bearing formula uses the Mercator factor:

Δψ = ln(tan(π/4 + φ2/2) ÷ tan(π/4 + φ1/2))

Where φ1 and φ2 are start and end latitudes in radians, and Δψ is the difference in “stretched” Mercator latitudes.

The rhumb bearing:

θ = atan2(ΔL, Δψ)

Where ΔL is the longitude difference.

The rhumb distance:

d = √(Δφ² + q² × ΔL²) × R

Where q = Δφ/Δψ (or cos(φ1) for nearly-equal latitudes) and R is Earth’s radius (6371 km).

Mercator projection and rhumb lines

Gerardus Mercator developed his famous projection in 1569 specifically to make rhumb lines appear as straight lines. This was revolutionary for ocean navigation:

• Before Mercator: navigators struggled to convert their straight-line compass courses to actual paths
• After Mercator: drew a straight line on the chart, measured the angle, set the compass to that angle
• Side effect: dramatic distortion of high-latitude areas (Greenland appears the size of Africa)

The Mercator projection is still the standard for marine navigation today, primarily because of the rhumb line property.

Rhumb line vs great circle, and the table that gets it wrong

The usual presentation looks like this:

That table is the one every reference prints, and it is misleading. Distance is not what drives the gap. What drives it is how much of the route runs east or west, and how far from the equator it does so. Run these six through the calculator above:

Anchorage to Oslo is the shortest passage in that list and by far the worst rhumb line, because it runs almost due east at 60°N. Lisbon to Rio is longer and costs essentially nothing, because it runs north-south. Both sit in the “2,500-5,000 nm, 5-10%” row of the first table, and neither is anywhere near 5-10%.

Two route shapes give exactly zero difference, and they are the boundary cases worth remembering. A course due north or due south follows a meridian, which is itself a great circle. A course along the equator is the same. Everything else falls somewhere between those and Anchorage to Oslo.

Real-world example: New York to London

Approximate coordinates: NYC at 40.7°N, 74.0°W. London at 51.5°N, 0.1°W.

Rhumb line:

• Distance: ~3,130 nm
• Bearing: 078° (east-northeast)
• Constant heading throughout
• Easy to navigate

Great circle:

• Distance: ~3,010 nm (about 120 nm shorter)
• Initial bearing: 052° (more northerly)
• Bearing changes constantly throughout voyage
• Maximum latitude: ~63°N (passes near Iceland)
• Requires course updates every hundreds of miles

The ~120 nm savings is about 4% less distance and fuel, and a several-hour-shorter trip. Worth the navigation complexity for fuel-conscious shipping.

Why rhumb lines spiral toward the poles

A constant-bearing path that is not due north, south, east or west spirals toward a pole, winding round it infinitely many times while covering a finite distance. The closer the course is to north-south, the tighter the approach; the closer to east-west, the more laps it takes.

Sail constantly at 350°, almost due north, and you will reach the North Pole. Sail at 349° and you spiral in to it as well, just with more turns.

Due west at 270° is the exception, and it is the one most often stated wrongly. A course of exactly 090° or 270° is a parallel of latitude, so you circle the Earth at that latitude forever and never get one meter closer to the pole. The spiral needs a north or south component to exist at all.

This is a mathematical curiosity. No voyage is long enough to demonstrate it, though it is the reason rhumb lines get unusable above about 60° latitude.

Great circle navigation in the age of GPS

Before GPS, great circle navigation required:

1. Compute multiple “waypoints” along the great circle
2. Sail rhumb line segments between waypoints
3. Update bearing at each waypoint
4. Adjust for prevailing winds and currents

Today, GPS computes the great circle automatically and constantly updates the recommended heading. The pilot just follows the GPS bearing.

When great circle isn’t optimal

Real-world routing rarely follows pure great circles:

Prevailing winds:

• Trade winds: better to sail with the wind, not against
• Westerlies in mid-latitudes: route to take advantage
• Doldrums (ITCZ): avoid by sailing further north or south

Ocean currents:

• Gulf Stream: ~3-5 knots eastward
• North Atlantic Drift: continues across Atlantic
• Plan routes to use favorable currents

Weather routing:

• Modern routing software optimizes for actual forecast conditions
• May suggest routes far from great circle
• Saves more time than pure great circle navigation

Examples of routed voyages:

• Eastbound transatlantic: typically uses Gulf Stream → North Atlantic Drift (composite of great circle + favorable current)
• Westbound transatlantic: typically routes south to avoid contrary current
• Pacific voyages: large variations based on wind systems and currents

Navigation in restricted waters

In coastal navigation, rhumb lines remain dominant:

• Chart navigation: drawing straight lines on Mercator chart
• Buoy-to-buoy passages: typically straight lines
• Channel transits: follow specific bearings
• Pilotage: distances too short for great circle to matter

Rhumb line math is built into all marine GPS units, chartplotters, and navigation software.

Aviation considerations

Aviation faces similar decisions:

• Short flights (under 500 nm): typically follow simple direct routes
• Medium flights: usually rhumb line or great circle depending on operator
• Long flights: great circle routing standard

Modern airliners use GPS-based RNAV (Area Navigation), which follows the great circle without intermediate waypoints.

The polar route problem

Great circles near the poles produce strange-looking routes on flat maps:

• Tokyo to New York: great circle goes over Alaska
• Sydney to Buenos Aires: great circle passes near Antarctica
• Los Angeles to Beijing: great circle passes near Aleutian Islands

These routes look bizarre on Mercator projections but are simply the shortest path on a sphere.

Practical navigation rules

For coastal cruising (typical sailboat use):

1. Use rhumb lines for daily passages
2. Plot on Mercator chart with parallel rules
3. Measure bearing with compass rose
4. Adjust for variation (magnetic vs true)
5. Note distance with dividers
6. Sail constant heading between waypoints

For ocean passages:

1. Plan great circle for overall route
2. Break into rhumb line segments of 500-1000 nm
3. Weather routing for actual conditions
4. Update plan as you progress

Cross-track error

When following any planned course (rhumb or great circle), navigators monitor cross-track error, meaning how far off the planned track they’ve drifted:

• Wind and current push the boat sideways
• Tides cause drift
• Compass errors compound

Acceptable cross-track error depends on situation:

• Open ocean: 1-5 nm acceptable
• Coastal: 0.5-1 nm acceptable
• Restricted waters: less than 0.1 nm
• Channel transit: must stay within channel marks

Common rhumb line mistakes

1. Using rhumb in polar regions: spiral becomes problematic above 60° latitude
2. Magnetic vs true confusion: forgetting to apply variation
3. Crossing the date line: longitude wraps around
4. Crossing the equator on N-S courses: bearing changes from N to S
5. Forgetting Earth isn’t flat: rhumb line is “straight” on chart but not on globe
6. Wrong projection chart: rhumb line only works on Mercator
7. Compass deviation: ignoring local magnetic interference
8. Not updating for current: drift accumulates over time
9. No GPS check: dead reckoning has compounding errors

On a coastal passage the rhumb line and the great circle are the same to within a rounding error, so steer the constant bearing and think no more about it. Past that, run the pair through the calculator before deciding, because the answer depends on which way the route runs and not on how long it is. A north-south leg costs nothing whatever its length; a long east-west leg at high latitude can cost a third of the distance. Modern GPS computes both automatically, but knowing which one you are following, and why, still saves fuel and heads off the classic “why does my heading keep drifting?” confusion.