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

The departure against length, on Web Mercator at 45°

Drawn here at the parameters it defaults to, with every essay that calls it.
The departure against length, on Web Mercator at 45°. six east–west segments from 50 to 1600 kilometres, each stored as two points and drawn straight, with the worst distance from the geodesic measured for each. On logarithmic axes the points lie on a line of slope 2.001 — the departure of a chord from an arc goes as κL²/8, so two is the prediction and the fit is the test of it. A segment of 50 km is 49 metres out; sixteen times the length is 1029 times the error rather than sixteen.

six east–west segments from 50 to 1600 kilometres, each stored as two points and drawn straight, with the worst distance from the geodesic measured for each. On logarithmic axes the points lie on a line of slope 2.001 — the departure of a chord from an arc goes as κL²/8, so two is the prediction and the fit is the test of it. A segment of 50 km is 49 metres out; sixteen times the length is 1029 times the error rather than sixteen.

It is drawn by operation-figure with show: "departure-scaling" — one member of a family of 14 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.

2 essays call 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.

London to Tokyo in three straight legs. The great circle, and the route a plan of three constant-heading legs actually follows between waypoints on it. The two touch at the waypoints and part between them by up to 456 km, and the flown route is 253 km longer than the direct one. The headings are 50°, 109°, 149°. Two routes. The great circle is 9559 km and is the shortest path on the sphere. The rhumb line holds a single compass bearing the whole way and is 11296 km — 1737 km further, or 18.2 per cent. On Orthographic the rhumb line departs from straight by 2.4e-1 of its own length. Paths and directions

Flying a curve in straight legs

Nobody steers a great circle, because a great circle requires the heading to change continuously. What is actually flown is a handful of constant-heading legs between waypoints on it, and the gap between plan and curve falls as the square of the number of legs.

 A straight line stored in Web Mercator, and where it really goes. Two points 5570 kilometres apart, joined by a straight segment in the plane the file's coordinates are in. Drawn on that plane it is a straight line and looks like the route; on the ground it is the curve marked measured, and the geodesic between the same two endpoints is the other one. The profile below is the distance between them along the line, reaching 718 kilometres at 49 per cent of the way across. Nothing here is an error in the data: both endpoints are exact, and the whole of the discrepancy is the word straight. What a machine does with it

A straight segment is a claim about a plane

Two exact endpoints, joined by a straight line in the plane the file is stored in. On the ground the line is 718 kilometres from the route it claims between New York and London, and 2,961 between London and Tokyo. The departure grows as the square of the length — fitted exponent 2.001 — so a stated tolerance costs vertices as a square root.

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