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

Weighted at their own size, a projection's pins move the shape further than the fit says, and further with every place

Drawn here at the parameters it defaults to, with every essay that calls it.
Weighted at their own size, a projection's pins move the shape further than the fit says, and further with every place. The places are fitted to every pairwise distance, each good to one per cent, and to a projection's positions as pins, with a shift, turn and size fitted jointly and each pin's tolerance set to the projection's own leftover error. The vertical axis is how far the projection's actual error moves the fitted shape, over how far the fit's covariance says its pins move it. Pins whose errors were independent would sit on the dashed line at one, and do: drawn that way forty times, they give 0.99. The regional Lambert conic goes from 1.38 at 8 places to 4.30 at 128; Mollweide from 1.52 to 4.05; Mercator from 2.95 to 4.76.

The places are fitted to every pairwise distance, each good to one per cent, and to a projection's positions as pins, with a shift, turn and size fitted jointly and each pin's tolerance set to the projection's own leftover error. The vertical axis is how far the projection's actual error moves the fitted shape, over how far the fit's covariance says its pins move it. Pins whose errors were independent would sit on the dashed line at one, and do: drawn that way forty times, they give 0.99. The regional Lambert conic goes from 1.38 at 8 places to 4.30 at 128; Mollweide from 1.52 to 4.05; Mercator from 2.95 to 4.76.

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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