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

A straight line stored in Web Mercator, and where it really goes

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

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*.

It is drawn by operation-figure with show: "segment-departure" — 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.

4 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.

 Three curves between 45°N and 55°N, 3026 km apart. The normal section observed from the first point, the normal section observed from the second, and the geodesic, each plotted as its distance to one side of the great-circle chord between the two ends. The two sections are 79.0 metres apart at their widest and the geodesic runs between them, 53.3 metres from the first. In LENGTH they differ by almost nothing — 2.40 millimetres over 3026 kilometres — so an observation that follows the wrong one measures the right distance along the wrong ground. On a sphere all three coincide exactly. What is taught wrongly

The normal section is not the geodesic

A theodolite at A sighted on B swings in one plane and the instrument at B sighted back swings in another, so the two observations trace different curves on the ground — 79 metres apart over 3,026 kilometres — and the shortest path is neither of them. The lengths differ by 2.4 millimetres, so the wrong curve measures the right distance along the wrong ground.

 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.

 Two readings of the same triangle, and the band between them. The same three vertices, joined two ways: with straight lines in the plane the coordinates are stored in, and along the ground. Every dot is a point the two readings disagree about — inside on one and outside on the other. The band covers 29.5 per cent of the polygon, which is 4241 thousand square kilometres, and it is not an error in either reading: the file does not say which one it means. Densifying the stored boundary removes it, which is the only fix there is. Drawn in Equirectangular. What a machine does with it

Inside is a claim about the edges

Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.

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