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

The best flat picture of them, and what it still gets wrong

Drawn here at the parameters it defaults to, with every essay that calls it.
The best flat picture of them, and what it still gets wrong. The four places placed on a plane so that the WORST of the six distances is as nearly right as any flat arrangement can make it. There is no projection here and no map: the dots carry no coordinates, only separations. Each edge is labelled with how far its drawn length is from the truth once the picture has been given its one free scale. The best any arrangement achieves is 0.629% on the worst edge — London to Tokyo — and no rearrangement lowers it, because the six numbers are not the distances of four points in a plane at all.

The four places placed on a plane so that the WORST of the six distances is as nearly right as any flat arrangement can make it. There is no projection here and no map: the dots carry no coordinates, only separations. Each edge is labelled with how far its drawn length is from the truth once the picture has been given its one free scale. The best any arrangement achieves is 0.629% on the worst edge — London to Tokyo — and no rearrangement lowers it, because the six numbers are not the distances of four points in a plane at all.

It is drawn by curvature-figure with show: "distance-flat" — 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 four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes. The impossibility

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

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