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 3 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 twelve rungs of this ladder have taken to be the curvature of the thing being mapped.

Where the surface curves the other way

Twelve rungs argue the impossibility 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 -1.60 at a coefficient of determination of 0.913. The previous rung's level-set measurement gave −1.33 and the argument predicts −1, so measuring the height directly moves the answer FURTHER from the prediction rather than towards it.

The height of the pass between two basins

The previous rung recorded a shortfall: the fracture threshold falls as the −1.33 power where the argument predicts −1, and the difference was supposed to be the height of the pass. It is not. Measuring the pass directly gives −1.60, which is further from the prediction, and the two basins turn out to be exact symmetric copies with nothing to decompose.

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.

HessianVerificationAspect searchBasinConformalityCoordinate independenceCritical pointCurvatureDevelopableEuler characteristicExponentFlat enough

All concepts