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

Weighted as they deserve, more pins never make the shape worse

Drawn here at the parameters it defaults to, with every essay that calls it.
Weighted as they deserve, more pins never make the shape worse. The root-mean-square error in each city's position relative to the picture's centre, against the pins' tolerance. With one place pinned it is 63.0 km at every tolerance, all of it from the distances and bearings. With two places pinned it falls to 57.3 km when the pins are good to a kilometre; with three, to 40.8; with four, to 1.2, because four pins good to a kilometre are a better drawing of the cities than the measurements between them. At fifty kilometres the four figures are 63.0, 61.1, 53.5, 42.6 km, and every curve returns to 63.0 as the pins become useless.

The root-mean-square error in each city's position relative to the picture's centre, against the pins' tolerance. With one place pinned it is 63.0 km at every tolerance, all of it from the distances and bearings. With two places pinned it falls to 57.3 km when the pins are good to a kilometre; with three, to 40.8; with four, to 1.2, because four pins good to a kilometre are a better drawing of the cities than the measurements between them. At fifty kilometres the four figures are 63.0, 61.1, 53.5, 42.6 km, and every curve returns to 63.0 as the pins become useless.

It is one of 78 figures drawn by the same construction, and the drawing above is this one with nothing changed.

1 essay calls 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.

Pins at one place move where the picture sits; pins at four reshape it. The best flat picture of four cities, fitted to every distance with a tolerance of 1 per cent, one bearing a pair to 0.01 radians, and pins to the sheet good to 50 km. Each ellipse is the error in a city's position relative to the picture's centre — its shape, which no shift changes — drawn 30 times its size. Solid: the part the distances and bearings put there. Dashed: the part the pins put there. Pinned at London alone, the pins put 0.00 km into the shape and the places 63.0 km. Pinned at all four, the pins put 30.5 km and the places 29.7 km, and the shape as a whole is 42.6 km against 63.0. The impossibility

A second pin is a measurement of the places

Counted exactly, a picture of places is a direct sum: the distances and bearings spend its shape and two pins spend where it sits on the sheet. Given tolerances, the sum survives in one form and fails in another. Pins at one place put none of their error into the shape at any tolerance, and wherever the pins are, their average takes none of the distances' error. But every place pinned after the first is a statement about the geography, and held exact it makes the shape worse once its error passes about two thirds of the shape's own.

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