Great Circle Routes

Pick two places on the globe. The page flies the shortest route between them beside the route that keeps one compass heading, prints both headings and both lengths, and splits the route into legs the way a navigator would.

Worked example — Great Circle Routes

Great circle, the shortest route	8,760.1 km
Heading on leaving A	311.7°
Heading on arriving at B	214.1°
Furthest the route gets from the equator	62.26° N
Change in Mercator height (isometric latitude)	−0.4208
The one heading that reaches B	258.4°
Length on that one heading	9,732.5 km
One heading, longer than the great circle by	972.4 km

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Great Circle Routes

A great circle is where a plane through the centre of a sphere cuts its surface. Its shorter arc between two points is the shortest path between them on the sphere, the sphere's version of a straight line. A rhumb line, or loxodrome, crosses every meridian at the same angle, so it can be steered on one compass heading; on Mercator's map it is a straight line. Unless both places are on the equator or on one meridian, the great circle is shorter and its heading changes all the way along it.

What each symbol means

σ: the angle between the two places seen from the centre of the Earth; the great circle is Rσ long, with σ in radians
θ: a heading, measured clockwise from north: θ₁ leaving A, θ₂ arriving at B
ψ: isometric latitude, ln tan(45° + φ/2): the height of latitude φ on Mercator's map of a unit globe
R: the Earth's radius, here the mean radius 6,371.0088 km

Assumes: A sphere of the Earth's mean radius. The Earth is flattened by about 1 part in 298, and on the WGS 84 ellipsoid the shortest path from London Heathrow to Los Angeles is 8,781.1 km by Vincenty's method, 0.24% longer than the sphere's 8,760.1 km. Over a sweep of routes the sphere is at worst about half a percent out, on short north–south routes near the equator.

To reach a city south-west of you, set off north-west 🖖

Press London to Los Angeles. Los Angeles is 17.5 degrees of latitude south of London and 118 degrees west, and a navigator who held one heading, 258.4°, would get there in 9,732.5 km. The shortest route leaves on 311.7°, north-west, crosses southern Greenland at 62.26° N and comes down into Los Angeles on 214.1°, heading south-west: 8,760.1 km, 972.4 km less. Watch the cyan needle on the compass. It swings through 97.5° while the route never bends left or right, because what moves is north: every meridian runs into the same pole, so a great circle crosses each one at a different angle. On the flat map below, the straight line is the one-heading route, and the great circle bows towards the pole to save nearly a thousand kilometres.

A parallel of latitude is not a straight line 🖖

Madrid is at 40.49° N and New York at 40.64° N, so the one-heading route between them runs almost due west, 270.2°, and almost along the parallel: 5,930.8 km. The great circle leaves Madrid on 294.7°, crosses the Atlantic up at 46.30° N and arrives 169.9 km sooner. Walking a parallel feels straight and is not. Away from the equator a parallel is a small circle drawn round the pole, and to stay on 40.49° N you would have to turn towards the pole by 0.768° every 100 km; let go of the wheel and you drift south of it along a great circle. The equator is the only parallel that needs no turning, because it is the only one whose centre is the centre of the Earth.

Halve the legs and the extra falls to a quarter, until the route passes near a pole 🖖

Ships followed great circles long before autopilots, as a few legs on fixed headings between waypoints. Press Four legs: London to Los Angeles on 299.2°, 267.1°, 236.5° and 219.3° is 74.6 km longer than the great circle, against 972.4 km for one heading. The table runs the halving, 972.4, 282.0, 74.6, 18.9, 4.8 km, and the ratio settles on 4. The reason is that a short leg's extra length grows as the cube of the leg: halve the leg and each one costs an eighth as much, with twice as many of them. Press Singapore to New York and the rule fails. That route passes 2.53° from the pole, where a fixed heading swings round the pole instead of cutting across it, and two legs still cost 2,976.7 km extra against 3,160.4 for one; the ratios wander from 1.06 up to 3.92 and back to 2.27, and only from 32 legs on, each about 480 km, is the ratio back at 4.

Problem solved in full

Hold one heading for ever 4 steps
Leave a point on the equator on a heading of 45°, north-east, and never change it. How far do you go before you reach the North Pole, and how many times do you circle it on the way? Press London to Los Angeles first: the page works out that rhumb line's length, and the same rule answers the question.

Open this problem in the tool
1. The page's length for the one-heading route from London to Los Angeles, split into a north–south part and an east–west part, both measured as angles at the Earth's centre, in radians:
  
  $d_{r} = R Δ φ^{2} + (q Δ λ)^{2} = 6371.0088 \times 0.30595 6^{2} + 1.49667 8^{2} = 9, 732.5 km$
2. Along a rhumb the heading never changes, so every kilometre gains the same northing, cos θ of a kilometre. From the equator to the pole is a quarter of a meridian, πR/2:
  
  $d = \frac{π R /2}{cos 4 5 ^{\circ}} = \frac{10 , 007 . 56}{0 . 707107} = 14, 152.8 km$
3. The longitude is the other half. On a rhumb, tan θ = Δλ/Δψ, so at 45° the longitude gained equals the change in isometric latitude ψ, in radians, and each 2π of it is one circuit of the pole. One degree short of the pole, 157.3 km of spiral from the end:
  
  $Δ λ = tan 4 5^{\circ} Δ ψ = ψ (8 9^{\circ}) = 4.7413 \Rightarrow n = \frac{4 . 7413}{2 π} = 0.755$
4. But ψ never stops growing. Close to the pole ψ is very nearly the logarithm of 2 over the remaining colatitude, so every tenfold step closer adds the same amount to it, and the same share of a turn:
  
  $ψ (φ) \approx ln \frac{2}{π /2 - φ} \Rightarrow Δ n = \frac{ln 10}{2 π} = 0.3665$
Answer

14,152.8 km, and infinitely many turns. The distance is finite because each kilometre of spiral gains 0.707 km of northing, so the whole spiral is √2 times a quarter meridian. The turns never end because ψ has no ceiling, but they come late and small: 0.755 of a turn by 89° N, 1.488 by 89.99° N with 1.573 km of spiral left, and from there another 0.3665 of a turn for every tenfold step closer. Mercator's map cannot show the end of it, because the pole is infinitely far up the sheet.

$d = 14, 152.8 km, n \to \infty, n (89.9 9^{\circ}) = 1.488$

Learning path

Geometry on a curved surface

References (2)

Geodesics on the Earth: the great circle as the sphere's shortest path, Clairaut's relation sin α₀ = sin α cos β that fixes how the heading changes along one (on a sphere, the sine rule), and the WGS 84 semi-axes this page's mean radius (2a + b)/3 is taken from: C. F. F. Karney, "Algorithms for geodesics", Journal of Geodesy 87(1), 43–55 (2013).
The ellipsoidal distance the lesson compares the sphere with: Vincenty's iterative inverse formula, used here on the WGS 84 ellipsoid for London Heathrow to Los Angeles (8,781.1 km against the sphere's 8,760.1 km): T. Vincenty, "Direct and inverse solutions of geodesics on the ellipsoid with application of nested equations", Survey Review 23(176), 88–93 (1975).

By the 42Math team · Reviewed against published sources. Corrections welcome.