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.

For most short voyages (under 1,000 miles), the difference is negligible. For long ocean crossings, choosing the wrong type costs days and fuel.

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 — but locally, for 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 — when each matters

For typical voyages:

For trans-Pacific or trans-Atlantic crossings, great circle saves hundreds of miles — significant in fuel costs and time.

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: ~3,640 nm
• Bearing: 073° (east-northeast)
• Constant heading throughout
• Easy to navigate

Great circle:

• Distance: ~3,470 nm (170 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 170 nm savings translates to about 5-8% less fuel and about half a day shorter trip. Worth the navigation complexity for fuel-conscious shipping.

Why rhumb lines spiral toward the poles

A constant-bearing path that isn’t due north, south, east, or west spirals toward a pole. The closer to N-S, the steeper the spiral; the closer to E-W, the gentler.

If you sail constantly at 350° (almost due north), you’ll reach the North Pole. If you sail at 270° (due west) at, say, 40°N, you’ll spiral closer to the pole but never reach it (the spiral asymptotes).

This is a mathematical curiosity. In practice, sailors never make voyages long enough to demonstrate the spiral.

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) — follow great circle naturally 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” — 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

Bottom line

A rhumb line (loxodrome) is a path of constant compass bearing — appears as a straight line on Mercator charts. Bearing: atan2(ΔLongitude, ΔMercator latitude). Distance is slightly longer than great circle. For voyages under 1,000 nm, rhumb and great circle are essentially the same. For trans-oceanic voyages, great circle saves 5-20% distance. Modern GPS automates both calculations. In practice, rhumb lines are still used for short coastal navigation while ocean passages combine great circles with weather routing for optimal results. Mercator’s 1569 projection was specifically designed to make rhumb lines straight lines — still the standard for marine charts today.