Figure

Up to a span, the best picture sits exactly on the floor

Drawn here at the parameters it defaults to, with every essay that calls it.

The four cities shrunk about their centre, with the worst bearing error of the best flat picture, the floor that no picture can go under, and Mercator's layout, against the span of the set. At every span up to 56° the best picture reaches the floor to the arithmetic — 77.22° at 56°, 39.52° at 40°, 13.56° at 20°, 5.68° at 10°. By 71° it has lifted clear of it and stays there, at 116° to 118° against floors of 64° to 70°.

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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