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Great-Circle vs Rhumb Line: Why Flight Paths Look Curved on Flat Maps

2026-03-01 · cartography

Great-Circle vs Rhumb Line: Why Flight Paths Look Curved on Flat Maps

TL;DR

When flights look "bent" on world maps, that is usually correct, not weird.

So the curved line on your map can be the shorter route in the real 3D world.

1) The core geometry in one picture (mental model)

Imagine Earth as a globe, then slice it with a plane that passes through Earth’s center and both cities.

A rhumb line is different:

2) Why projection creates the visual illusion

Most web maps use Web Mercator-like views for interaction and consistency.

That is why a polar-leaning transoceanic flight often looks like an upward arch on screen.

3) Math you can reuse

For two points ((\phi_1, \lambda_1)), ((\phi_2, \lambda_2)) on a spherical Earth radius (R):

Great-circle (haversine form)

[ a = \sin^2\left(\frac{\Delta\phi}{2}\right) + \cos\phi_1\cos\phi_2\sin^2\left(\frac{\Delta\lambda}{2}\right) ] [ \delta = 2\arcsin(\sqrt{a}), \quad d_{gc}=R\delta ]

Rhumb-line distance (spherical)

[ \Delta\psi = \ln\left(\frac{\tan(\pi/4 + \phi_2/2)}{\tan(\pi/4 + \phi_1/2)}\right), \quad q = \begin{cases} \frac{\Delta\phi}{\Delta\psi}, & |\Delta\psi|>\epsilon \ \cos\phi_1, & \text{otherwise} \end{cases} ] [ d_{rhumb} = R\sqrt{(\Delta\phi)^2 + q^2(\Delta\lambda)^2} ]

(Use anti-meridian-safe (\Delta\lambda) normalization to pick the shorter longitudinal wrap.)

4) Concrete route comparisons (sphere approximation)

Using mean Earth radius (R=6371.0088,km):

Route Great-circle (km) Rhumb (km) Extra km (rhumb - gc) Extra %
ICN → JFK 11,089.9 13,798.0 2,708.1 24.42%
ICN → LHR 8,861.3 10,116.8 1,255.5 14.17%
LAX → NRT 8,752.9 9,235.3 482.4 5.51%
SIN → SFO 13,580.1 14,291.0 710.9 5.23%
SYD → SCL 11,346.7 12,783.6 1,436.9 12.66%

Takeaway: the penalty of constant-bearing routing can be modest or huge depending on latitude and longitude separation.

5) Operational intuition (aviation vs old marine navigation)

Why great-circle dominates modern flight planning

Why rhumb lines were historically attractive at sea

6) Common misunderstandings to avoid

  1. "Straight on map = shortest"

    • Only true in some projections/situations. On Mercator, it is often false for long-haul routes.

  2. "Great-circle means one fixed heading"

    • No. Initial and final headings can differ substantially.

  3. "Any curved map line is geodesic"

    • Not necessarily. Many consumer maps render smoothed or stylized paths.

  4. "Sphere math is exact"

    • Earth is ellipsoidal. Sphere formulas are usually close (often within about 1%) but not exact.

7) Tiny Python check

import math

R = 6371.0088  # km

def great_circle_km(lat1, lon1, lat2, lon2):
    p1, p2 = math.radians(lat1), math.radians(lat2)
    l1, l2 = math.radians(lon1), math.radians(lon2)
    dp, dl = p2 - p1, l2 - l1
    a = math.sin(dp/2)**2 + math.cos(p1)*math.cos(p2)*math.sin(dl/2)**2
    return R * (2 * math.asin(math.sqrt(a)))

def rhumb_km(lat1, lon1, lat2, lon2):
    p1, p2 = math.radians(lat1), math.radians(lat2)
    l1, l2 = math.radians(lon1), math.radians(lon2)
    dp, dl = p2 - p1, l2 - l1

    # anti-meridian-safe delta-lon
    if abs(dl) > math.pi:
        dl = dl - 2*math.pi if dl > 0 else dl + 2*math.pi

    dpsi = math.log(math.tan(math.pi/4 + p2/2) / math.tan(math.pi/4 + p1/2))
    q = dp / dpsi if abs(dpsi) > 1e-12 else math.cos(p1)

    return R * math.hypot(dp, q*dl)

8) Practical rule-of-thumb

If your route is long and especially if it spans higher latitudes, assume:

For product work (maps, tracking, logistics UIs), always label whether drawn segments are:

That one label prevents a lot of user confusion.

References