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

What a loop round a parallel turns a direction by, and what it encloses

Drawn here at the parameters it defaults to, with every essay that calls it.
What a loop round a parallel turns a direction by, and what it encloses. Two quantities against the latitude of the loop. The turning — the angle a transported direction comes back rotated by, relative to the local north — is 2π sin φ, measured. The curvature the loop encloses is the area of the cap above it, 2π(1 − sin φ). They sum to 2π at every latitude, drawn as the flat line across the top, and neither is the other: at 40° the turning is 4.039 radians and the enclosed curvature 2.244. The difference is the winding of "north" itself around the pole, which is exactly one revolution however small the loop.

Two quantities against the latitude of the loop. The turning — the angle a transported direction comes back rotated by, relative to the local north — is 2π sin φ, measured. The curvature the loop encloses is the area of the cap above it, 2π(1 − sin φ). They sum to 2π at every latitude, drawn as the flat line across the top, and neither is the other: at 40° the turning is 4.039 radians and the enclosed curvature 2.244. The difference is the winding of "north" itself around the pole, which is exactly one revolution however small the loop.

It is one of 78 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.

A direction carried once round the sphere. A vector transported round a closed loop on the sphere, kept as parallel to itself as the surface allows at every step — the component that leaves the tangent plane is removed and nothing else is done to it. The heavy arrows are its direction at the start and at the finish, drawn from the same point; the light ones are its direction along the way. It comes back turned through 282.0 degrees, which is 0.783 of a revolution, and nothing in the transport turned it. Drawn in an orthographic projection of the embedding, which is itself a map and has its own distortion. The impossibility

A direction carried round a loop

Carry a bearing round a circuit, keeping it as parallel to itself as the surface allows, and it comes back turned. Round a parallel at 45° the turn is 4.443 radians and the cap enclosed is 1.840, and they sum to exactly one revolution — so the turning is not the curvature, and on a cone it is all of one and none of the other.

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