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

Past a few dozen places, a projection's pins weighted at their own size make the shape worse than none

Drawn here at the parameters it defaults to, with every essay that calls it.
Past a few dozen places, a projection's pins weighted at their own size make the shape worse than none. The error in the fitted shape with a projection's positions added as pins, each weighted at the projection's own leftover error, divided by the error the distances leave alone. Below the dashed line the pins help. With every pair of places measured, the regional Lambert conic halves the error at 8 places (0.48) and makes it worse from 24 (1.12), reaching 1.50 at 128. With each place measured only to its six nearest, the conic still helps until about a hundred places, and Mollweide's pins — weighted, again, at exactly their own size — make the shape up to 1.81 times as wrong as the distances alone, at 48 places.

The error in the fitted shape with a projection's positions added as pins, each weighted at the projection's own leftover error, divided by the error the distances leave alone. Below the dashed line the pins help. With every pair of places measured, the regional Lambert conic halves the error at 8 places (0.48) and makes it worse from 24 (1.12), reaching 1.50 at 128. With each place measured only to its six nearest, the conic still helps until about a hundred places, and Mollweide's pins — weighted, again, at exactly their own size — make the shape up to 1.81 times as wrong as the distances alone, at 48 places.

It is one of 86 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.

Mercator's positions for twenty-four cities are mostly a size, and the rest is one pattern. Twenty-four European cities from Lisbon to Moscow. Solid dots: the best flat picture of their 276 great-circle distances, north up at 15°E 50°N. Lines run from each dot to where Mercator puts the city. Left, with only the sheet's origin moved: the projection is 662 km off in root-mean-square, because it draws the region 1.56 times too large. Right, after the four numbers any registration fits for itself — a shift, a turn of 1.73° and that size — what is left is 154 km, drawn 4 times its length: 5.4 per cent of the squared error, and it is a smooth swirl rather than an error at each city. The impossibility

A projection pins every place, and its error does not average

A projection used as a stated position for every place in a region splits the way a set of pins does: a shift, a turn and a size that any registration takes for itself, and a pattern that competes with the geography. Over twenty-four European cities 94 per cent of Mercator's error is size. What is left is one smooth displacement, so a fit told its size but not its shape says the pins move the picture as little as a quarter as far as they do — and with the weight a pin of that size deserves, past a few dozen places the projection makes the shape worse than no projection at all.

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