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

A cubic registration takes the pattern out, and a quadratic one can make a world map worse

Drawn here at the parameters it defaults to, with every essay that calls it.
A cubic registration takes the pattern out, and a quadratic one can make a world map worse. The same sixty-four places and six-nearest network, with the registration between each projection's positions and the fitted places allowed more freedom: a shift, turn and size (four numbers), an affine map (six), a quadratic (twelve), a cubic (twenty). Solid: the actual error in the shape. Dashed: what the fit states. Distances alone: 8.50 km. Under a cubic the regional conic gives 3.99 km against a stated 3.95 km, where under the four-number registration it gave 6.21 km against 3.69 km. Mercator under a quadratic is the worst point on the chart, 12.78 km against 9.38 km under four numbers: the quadratic shrinks its leftover from 184 km to 35 km, the fit trusts the pins accordingly, and what is left is still one pattern.

The same sixty-four places and six-nearest network, with the registration between each projection's positions and the fitted places allowed more freedom: a shift, turn and size (four numbers), an affine map (six), a quadratic (twelve), a cubic (twenty). Solid: the actual error in the shape. Dashed: what the fit states. Distances alone: 8.50 km. Under a cubic the regional conic gives 3.99 km against a stated 3.95 km, where under the four-number registration it gave 6.21 km against 3.69 km. Mercator under a quadratic is the worst point on the chart, 12.78 km against 9.38 km under four numbers: the quadratic shrinks its leftover from 184 km to 35 km, the fit trusts the pins accordingly, and what is left is still one pattern.

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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