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

The pins' share of the shape is nothing at one place, and grows with every place pinned

Drawn here at the parameters it defaults to, with every essay that calls it.
The pins' share of the shape is nothing at one place, and grows with every place pinned. How much of the error in the picture's shape comes from the pins, as a share of its variance, against the pins' tolerance, for pins at one to four of the four cities. At one place it is exactly nothing at every tolerance. At two places it peaks at 4.6 per cent near 40 km; at three, 20.7 per cent near 40 km. At four, pins good to a kilometre supply 100 per cent of the shape's error, pins good to fifty 51 per cent and pins good to three hundred 3 per cent. As the pins become useless every curve falls to nothing, because the fit stops listening to them. As they become exact the curves for two and three places fall to nothing too, because an exact pin has no error to give, and the curve for four rises to all of it, because four exact pins leave the distances and bearings nothing to decide.

How much of the error in the picture's shape comes from the pins, as a share of its variance, against the pins' tolerance, for pins at one to four of the four cities. At one place it is exactly nothing at every tolerance. At two places it peaks at 4.6 per cent near 40 km; at three, 20.7 per cent near 40 km. At four, pins good to a kilometre supply 100 per cent of the shape's error, pins good to fifty 51 per cent and pins good to three hundred 3 per cent. As the pins become useless every curve falls to nothing, because the fit stops listening to them. As they become exact the curves for two and three places fall to nothing too, because an exact pin has no error to give, and the curve for four rises to all of it, because four exact pins leave the distances and bearings nothing to decide.

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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