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

Wherever the pins are, the picture's place on the sheet takes no error from the places

Drawn here at the parameters it defaults to, with every essay that calls it.
Wherever the pins are, the picture's place on the sheet takes no error from the places. The error in where the pinned cities sit on the sheet, on average, split by where it came from, with every pin good to 50 km. Pinned at one place, 70.7 km, all from the pins; at two, 50.0; at three, 40.8; at four, 35.4 — the square root of two times the pins' tolerance, divided by the square root of how many places are pinned. The part from the distances and bearings is zero in every case, to rounding. Fifteen hundred seeded fits at full nonlinearity with two pinned places put the registration 49.8 km off and the shape 61.8 km off, against 50.0 and 61.1 computed.

The error in where the pinned cities sit on the sheet, on average, split by where it came from, with every pin good to 50 km. Pinned at one place, 70.7 km, all from the pins; at two, 50.0; at three, 40.8; at four, 35.4 — the square root of two times the pins' tolerance, divided by the square root of how many places are pinned. The part from the distances and bearings is zero in every case, to rounding. Fifteen hundred seeded fits at full nonlinearity with two pinned places put the registration 49.8 km off and the shape 61.8 km off, against 50.0 and 61.1 computed.

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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