Cartographic Projection
every map is wrong, and by exactly how much
Concept

Saddle — where it appears

A place where a surface curves up in one direction and down in another, so its Gaussian curvature is negative. It is the pass between two summits, and it is what most of a real landscape is: even a single smooth hill is a saddle over the whole of its skirt.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Where a ridge, a valley and the pass between them curves the other way. The Gaussian curvature of a stated terrain over a 80-kilometre window, with the sign shown by the colour and the size by the ink. Positive on the summits and in the hollows, negative everywhere between — which is most of it: 79 per cent of the curved area. The extremes are 2.2e+4 times the Earth's own curvature, which is the quantity every argument on a sphere takes to be the curvature of the thing being mapped.

Where the surface curves the other way

The impossibility of a perfect map is usually argued on surfaces whose curvature is positive everywhere. The surface a map of the ground actually depicts is not one of them: a stated terrain is saddle-shaped over 79 per cent of its curved area, its curvature runs to twenty-two thousand times the Earth's own, and even a single smooth hill is concave over 89 per cent of itself.

impossibility · Curvature The pass between the basins, measured rather than bracketed. The height of the lowest path from one near-optimal basin to the other, for regions of growing size. It is found by sorting the score surface and joining cells in order, so it is the minimax path's own height rather than the level at which a bisection stops finding two pieces. The fitted exponent is -1.34 at a coefficient of determination of 0.986. The earlier level-set measurement on the same projection gave -1.57 and the argument predicts −1, so measuring the height directly moves the answer towards the prediction, though not onto it.

The height of the pass between two basins

One thing was left owed: the fracture threshold falls faster than the argument predicts, and the difference was supposed to be the level set's blindness to the height of the pass. Measured directly, the pass between the two deepest basins falls as the −1.34 power at R-squared 0.986 — nearer the predicted −1 than the level set's −1.57 on the same projection, and not onto it. It is not the pass that breaks the set first, and at five of six sizes the second basin is the first one's mirror twin.

choosing · Choosing Twelve cells of the grid's pole row are one map. The pole row of the aspect grid for Robinson over a 10° square region centred at 38° N: 24 pole longitudes across, 12 turns of the third angle down, with the projection's pole on the south geographic pole. There, moving the pole in longitude and turning the third angle are the same rotation, so every cell whose longitude less its turn is the same is the same map. The twelve filled cells are one map, the lowest on the row at 1.71 times the region's best score, and they score alike to 2e-11. No two of them are neighbours. The hollow cells are the maps one 15° step round the axis, the lowest at 2.33 times the best, and on the grid as it was the copies could join only through them.

The pass was a map meeting itself

Four measurements fitted a power law to the depth of the pass that splits a near-optimal set of aspects, and found it tight — R-squared of 0.98 for Robinson, 0.996 along most of a family of maps. On the grid those aspects were searched over, a pole row holds every normal-aspect map twelve times at cells that are not neighbours, and in 28 of Robinson's 36 regions the published pass was two copies of one map meeting through the map one step round the axis. Joined as the one map they are, Robinson's curve falls to R-squared 0.008, and the sinusoidal's famous refusal turns out to be the one measurement the copies never reached.

choosing · Choosing The points a projection cannot move. The critical points of a stated field, found twice: once on the sphere from its own gradient, and once in Lambert cylindrical's page coordinates from the page's own numbers, with nothing shared between the two searches. seven points, 3 maxima, 2 saddles and 2 minima, and the two sets agree in position to 2.3e-7 degrees and in type at every one. The page's Jacobian is invertible wherever the map is a map, so it sends a zero gradient to a zero gradient and cannot move a critical point anywhere.

What the page cannot move

Six rungs measure what a page does to a field's readings and every one of them moves. The critical points do not: found on the sphere and found again in a projection's own page coordinates with nothing shared between the searches, they agree to 10⁻⁷ degrees and in type at every point, on every projection. What the page does move is their shape, by a factor bounded exactly by the indicatrix's axis ratio squared.

distortion · Gradient

Named alongside it

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

VerificationAspect searchBasinExponentHessianOptimisation landscapeReproducibilityAspectConformalityCoordinate independenceCritical pointCurvature

All concepts