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

The same places, arranged by a map and arranged by nothing

Drawn here at the parameters it defaults to, with every essay that calls it.
The same places, arranged by a map and arranged by nothing. On the left, London, New York, Tokyo, Sydney where Azimuthal equidistant puts them, centred on the set's own centroid. On the right, the arrangement that makes the worst of the six distances as nearly right as any flat arrangement can. The edges that carry more than half the worst error are drawn heavier. The projection reaches 8.28% and the free picture 0.63%. The free picture is better and is not a map: it has no graticule to hang a coastline on, nothing to say about any place not in the set, and adding a seventeenth city moves every dot in it.

On the left, London, New York, Tokyo, Sydney where Azimuthal equidistant puts them, centred on the set's own centroid. On the right, the arrangement that makes the worst of the six distances as nearly right as any flat arrangement can. The edges that carry more than half the worst error are drawn heavier. The projection reaches 8.28% and the free picture 0.63%. The free picture is better and is not a map: it has no graticule to hang a coastline on, nothing to say about any place not in the set, and adding a seventeenth city moves every dot in it.

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