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

What abandoning the projection is worth, and how fast it stops being worth it

Drawn here at the parameters it defaults to, with every essay that calls it.
What abandoning the projection is worth, and how fast it stops being worth it. For each set of places, the worst-edge error of the best projection in the library divided by the worst-edge error of the best flat arrangement of the same distances. The free picture cannot lose, because every projection's own layout is handed to the search as a starting point. On four world cities the freedom is worth a factor of 13.2; on sixteen it is worth 1.08, which is eight per cent. What collapses it is arithmetic rather than cartography: a picture of n places has 2n − 3 free numbers against n(n−1)/2 distances, so the freedom available per constraint falls as 4/n, and by sixteen places the optimiser has almost nothing left to spend that the projection has not already spent.

For each set of places, the worst-edge error of the best projection in the library divided by the worst-edge error of the best flat arrangement of the same distances. The free picture cannot lose, because every projection's own layout is handed to the search as a starting point. On four world cities the freedom is worth a factor of 13.2; on sixteen it is worth 1.08, which is eight per cent. What collapses it is arithmetic rather than cartography: a picture of n places has 2n − 3 free numbers against n(n−1)/2 distances, so the freedom available per constraint falls as 4/n, and by sixteen places the optimiser has almost nothing left to spend that the projection has not already spent.

It is drawn by curvature-figure with show: "distance-advantage" — 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.

Every projection against a picture that is not a map. The worst relative distance error over London, New York, Tokyo, Sydney, for each projection in the library and for the best flat arrangement of the same distances. Each azimuthal member is centred on the set's own centroid, which is the fair comparison. The free picture reaches 0.63% and the best projection, Azimuthal equidistant, reaches 8.28% — a ratio of 13.17. The free picture cannot lose, because every projection's own layout was handed to the search as a starting point; what the figure measures is how much the freedom is worth, and it is worth different amounts at different sizes. The impossibility

The best flat picture is not a map

A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.

The whole library