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Spherical great-circle and rhumb-line calculator
Compare great-circle and rhumb-line distances with an explicit sphere radius; inspect units, bearings, the dateline and polar limits.
navigation-sphere-vector/1.0.0
Great circle and rhumb line on a sphere
Compare two geometric tracks between invented coordinates. Choose the sphere radius explicitly; neither output is an ellipsoidal WGS84 solution or a checked passage route.
Method and symbols
- Latitude φ is north-positive; longitude λ is east-positive. Inputs are degrees; all trigonometric equations use radians. R is a user-declared spherical radius in metres. The default 6,371,008.8 m is an educational model choice.
- A great circle is a sphere’s shortest surface arc between non-antipodal points. Its forward bearing generally changes. A rhumb line crosses meridians at one constant true bearing.
- Select Δλ in (−π, π] to use the shorter longitude branch. Exactly 180° has equal eastward and westward choices; this implementation selects +π, eastward.
- δ is the great-circle central angle; Δφ = φ₂ − φ₁. ψ = ln tan(π/4 + φ/2) is the spherical Mercator latitude, which diverges at a pole. q is dimensionless.
- Metres = R × angular distance; kilometres = metres/1000; nautical miles = metres/1852, exactly. One degree is Rπ/(180 × 1852) NM, not a universal 60 NM.
u(φ, λ) = (cos φ cos λ, cos φ sin λ, sin φ)
δ = atan2(‖u₁ × u₂‖, u₁ · u₂); dGC = Rδ
C₁ = atan2(sin Δλ cos φ₂, cos φ₁ sin φ₂ − sin φ₁ cos φ₂ cos Δλ)
C₂ = reverse initial bearing + 180°, wrapped to [0°, 360°)
Δψ = ln[tan(π/4 + φ₂/2) / tan(π/4 + φ₁/2)]
q = Δφ/Δψ; when φ₁ = φ₂, q = cos φ₁
dRL = R √(Δφ² + q²Δλ²); CRL = atan2(Δλ, Δψ)
Current result
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Worked example · invented data
- Take two invented points A = (50° N, 30° W), B = (50° N, 30° E), on R = 6,371,008.8 m. They are geometry inputs, not ports or approved waypoints.
- φ₁ = φ₂ = 50π/180 ≈ 0.872664626 rad; Δφ = 0; Δλ = 60π/180 = π/3 ≈ 1.047197551 rad.
- For equal latitudes q = cos(50°) ≈ 0.642787610; Δψ = 0. Rhumb angular distance = qπ/3 ≈ 0.673125611 rad; bearing = 090°.
- For the great circle, cos δ = sin²(50°) + cos²(50°) cos(60°) ≈ 0.793412044. The stable cross/dot form gives δ ≈ 0.654402031 rad. The table below carries the calculated distances and both forward bearings.
- Multiply each angular distance by R for metres, then divide by 1000 or 1852. Compare the excess rhumb distance in metres; a shorter geometric line alone does not establish a usable sea route.
dGC = 6,371,010 m × 0.654402 = 4,169,200 m
dRL = 6,371,010 m × √[(0)² + (0.642788 × 1.0472)²] = 4,288,490 m
| Quantity | Value | Unit |
|---|---|---|
| Great circle · Distance | 4,169,200 | m |
| Great circle · Distance | 4,169.2 | km |
| Great circle · Distance | 2,251.19 | NM |
| Great circle · Initial true bearing | 66.1413 | ° true |
| Great circle · Final forward true bearing | 113.859 | ° true |
| Rhumb line · Distance | 4,288,490 | m |
| Rhumb line · Distance | 4,288.49 | km |
| Rhumb line · Distance | 2,315.6 | NM |
| Rhumb line · Constant true bearing | 90 | ° true |
| Rhumb distance minus great-circle distance | 119,288 | m |
| Quantity | Value | Unit |
|---|---|---|
| Δφ | 0 | rad |
| Δλ | 1.0472 | rad |
| δ | 0.654402 | rad |
| Δψ | 0 | 1 |
| q | 0.642788 | 1 |
Interpretation and limits
- This is a spherical calculation. It does not compute ellipsoidal geodesics, chart projection distortion, land avoidance, charted depths, traffic, weather, ice or route safety.
- Coincident points have zero distance and undefined bearings. At exact antipodes δ = π and infinitely many shortest great-circle arcs exist. Bearings are also suppressed within cross-product norm 10⁻¹² of an antipode.
- At an exact pole, this tool suppresses great-circle bearings because the local north/longitude reference is ambiguous. Great-circle distance remains available. Different longitudes at the same pole represent one point.
- The rhumb formula is available only for |φ| ≤ 89.999999°. Beyond this documented numerical boundary it returns unavailable, rather than pretending Mercator is finite at the pole. A warning begins above 89.9°. This domain rule also applies to coincident pole points.
- An exact 180° longitude difference has two equal-length rhumb branches. The selected eastward tie does not establish a unique safe direction. An antipodal great-circle distance may be compared to a rhumb distance, but no unique great-circle bearing is claimed.
- The implementation uses atan2 of vector cross/dot products for δ and a log1p identity for Mercator differences. For |Δφ| ≤ 10⁻⁸ min(cos φ₁, cos φ₂), it uses q ≈ cos((φ₁ + φ₂)/2) and Δψ ≈ Δφ/q to avoid dividing subnormal differences. CSV retains double precision; the interface shows up to six significant digits. Rounding can make a bearing display as 360° although it is stored below 360°.
Sources, method and version
Method: NGA Bowditch spherical sailing relationships; Ed Williams’s original Aviation Formulary rhumb equations adapted to east-positive longitude; GeographicLib author documentation for nonunique geodesics. This implementation is independently written and explicitly spherical.
navigation-sphere-vector/1.0.0 · verified 2026-10-09
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Related context
The method explanation and worked example are on this page. The articles below provide additional context.
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