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

The same arrangement at three sizes

Drawn here at the parameters it defaults to, with every essay that calls it.
The same arrangement at three sizes. The best flat picture of London, New York, Tokyo, Sydney, with the configuration shrunk about its own centroid so that every bearing from the centre is unchanged and only how much of the sphere it covers varies. At 152.8° across the worst edge is out by 0.629%; at 39.7° it is 0.0299%; at 9.9° it is 0.0018%. The pictures are indistinguishable to the eye and the numbers under them fall by a factor of sixteen for each factor of four, which is what a square law looks like when it is drawn rather than plotted.

The best flat picture of London, New York, Tokyo, Sydney, with the configuration shrunk about its own centroid so that every bearing from the centre is unchanged and only how much of the sphere it covers varies. At 152.8° across the worst edge is out by 0.629%; at 39.7° it is 0.0299%; at 9.9° it is 0.0018%. The pictures are indistinguishable to the eye and the numbers under them fall by a factor of sixteen for each factor of four, which is what a square law looks like when it is drawn rather than plotted.

It is drawn by curvature-figure with show: "distance-shrink" — one member of a family of 29 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

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.

The least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not. The impossibility

How wrong a flat picture has to be

The rung below proves no flat picture of four places is exact and leaves the size of the failure to a determinant nobody can read. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.

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