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

The rhumb line runs midway between the two bearings, to first order

Drawn here at the parameters it defaults to, with every essay that calls it.
The rhumb line runs midway between the two bearings, to first order. From London along a great circle at an initial bearing of 60°, to distances of 100 to 4,000 km. The upper line is how far the bearing out and the bearing back depart from being reverses of each other — 11.05° at 1,000 km — which grows in proportion to the distance (fitted slope 1.05). The lower line is how far the rhumb line's constant bearing sits from the mean of the two: 0.126° at 1,000 km, growing as the square of the distance (slope 1.98). So a straight line on Mercator splits a pair's two bearings evenly over any regional distance, which is what puts it on the floor.

From London along a great circle at an initial bearing of 60°, to distances of 100 to 4,000 km. The upper line is how far the bearing out and the bearing back depart from being reverses of each other — 11.05° at 1,000 km — which grows in proportion to the distance (fitted slope 1.05). The lower line is how far the rhumb line's constant bearing sits from the mean of the two: 0.126° at 1,000 km, growing as the square of the distance (slope 1.98). So a straight line on Mercator splits a pair's two bearings evenly over any regional distance, which is what puts it on the floor.

It is one of 44 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.

One line cannot point both ways between New York and Tokyo. The great-circle bearing from New York to Tokyo is 333.0°, and from Tokyo back to New York 25.1°. A flat picture draws one straight line between them, whose two directions differ by exactly 180°; the two bearings differ by 180° and another 127.9°. The dashed direction is the best a straight line can do for both ends at once, and it is 63.9° from the truth at each — which is a floor on the bearing error of every flat picture that contains these two cities. The impossibility

A flat picture has one direction between two places, and the Earth has two

The bearing from New York to Tokyo and the bearing back are not reverses of each other: they miss by 128 degrees. A flat picture draws one line between the two cities, so no picture containing both can have a bearing error below 64 degrees. On any set of places up to about fifty degrees across, the best flat picture sits exactly on that floor — and away from the poles, so does Mercator.

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