Concept

Cut locus — where it appears

The set of points at which a geodesic from a given point stops being the shortest route. On a sphere it is the antipode; on an oblate ellipsoid it is an arc of the antipodal meridian, sixty-six kilometres long for a point on the equator and shrinking to nothing at the pole.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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.

paths · Paths

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.

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.

paths · Paths

Everywhere within 200 km of the route from London to Tokyo. The shortest route between the two places, and the set of places within 200 kilometres of it, with both edges computed on the sphere and then projected. On the ground the set holds 3.949 million square kilometres — the band 3.823 and the two end caps 0.126, which between them make one disc of the corridor's own width. Drawn in Mollweide the corridor is visibly wider at one end than the other, and the ground it stands for is not.

A corridor has a width the page cannot keep

Buffering a line is the most-run operation in spatial analysis and the corridor it produces is a reach set with two failures nobody separates: stroked on the page, one width covers 190 to 471 kilometres of ground along a single route; computed in closed form, the formula stops being the area at a width the route's own length fixes, and eventually claims more ground than the sphere has.

paths · Reach

Every one of the cube's 384 nets, scored. All 384 spanning trees of the cube's face graph, unfolded and scored on the total separation their cuts leave: how far apart, in edge lengths, the two copies of each cut edge end up on the page. Every one of them is a valid net — no Platonic unfolding overlaps — and they range from 11.30 to 17.97, a factor of 1.59. The heuristic of unfolding outwards from a chosen face lands on the first of them, exactly.

What the net heuristic cannot find

Rung seven chose a net by unfolding outwards from a face and showed the choice beats guessing, and recorded that its own limit was unknown. Enumerated in full, the heuristic turns out to be exactly optimal on every solid small enough to check — rank one of 384 — and above that ceiling no sample can tell whether it still is, because eight hundred random nets never reach it.

families · Polyhedral

Named alongside it

The objects these essays reach for when they reach for this one.

Great circle Antipode Convergence Ellipsoid Flattening Geodesic Area Boundary Buffer Certificate Closed form Combinatorics

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