Figure

eight shortest routes from London to its antipode

Drawn here at the parameters it defaults to, with every essay that calls it.

Great circles leaving London on eight different bearings, every one of them arriving at the same point on the far side of the world after exactly the same 20015 km. Between a point and its antipode there is no shortest path, because there are infinitely many and they are all shortest. Displace the destination by 0.4° — 52 km — and one of them wins, but only just, which is the condition an iterative solver cannot handle.

It is drawn by route-figure with show: "antipodal-routes" — one member of a family of 9 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

3 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

London to Tokyo on Orthographic. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.9e-1 of its own length.

Paths and directions

The great-circle vertex

One quantity is constant along a shortest path on a sphere, and it fixes the highest latitude that path will reach before the journey starts. That number is why polar routes exist, and it can be read off the departure bearing without tracing the route at all.

Where the inverse problem stops converging, 3° around the antipode of London. Every target in a 6° square centred on the point diametrically opposite London, shaded by how many iterations Vincenty's inverse formula needed to find the geodesic to it. 3 of 625 — 0% — never converged at all, and the worst success took 113 steps against a handful anywhere else on Earth. The failure is not a defect in the formula: near the antipode the shortest path is nearly ambiguous, and an iteration looking for one answer is being asked which of many.

Paths and directions

The route with no shortest path

Between a point and the point diametrically opposite there are infinitely many shortest routes and no shortest route, and the standard formula for the distance between two places stops converging in a neighbourhood of it. The failure is a property of the question rather than a defect in the answer.

Where two equal geodesics meet, from 15° north. A geodesic leaving at azimuth α and its mirror image at −α have the same length wherever they meet, and by symmetry they meet on the antipodal meridian. On a sphere they all meet at one point, the antipode, to 1.4e-7° — the flat line. On the ellipsoid the meeting latitude moves with the azimuth, from -15.563° to -15.009°, so the set of points with two shortest routes is an arc 61 km long, and the distance to it varies by 31 km along its own length.

Paths and directions

Where the shortest route stops being the only one

On a sphere there is exactly one point with no shortest route from a given place: the antipode. On the ellipsoid the Earth actually is, that point is an arc — sixty-six kilometres of the antipodal meridian for a point on the equator, half a kilometre for one at 85°, and every point of it reachable by two different geodesics of exactly equal length.

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