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

Where longitude and latitude can be exchanged without noticing, to 1 km

Drawn here at the parameters it defaults to, with every essay that calls it.
Where longitude and latitude can be exchanged without noticing, to 1 km. An equirectangular frame with the line λ = φ drawn on it. Reversing the two numbers of a coordinate is the most common coordinate mistake there is and is usually called obvious; it is obvious everywhere except along this line, where it does exactly nothing. The marked points are 0°E 60°N, 40°E 40°N, 70°W 40°N, displaced by 8390 km, 0 km, 14872 km. The band inside which the mistake stays under 1 kilometre is 0.013° wide at 10° and 0.016° at 60° — narrow, and not zero.

An equirectangular frame with the line λ = φ drawn on it. Reversing the two numbers of a coordinate is the most common coordinate mistake there is and is usually called obvious; it is obvious everywhere except along this line, where it does exactly nothing. The marked points are 0°E 60°N, 40°E 40°N, 70°W 40°N, displaced by 8390 km, 0 km, 14872 km. The band inside which the mistake stays under 1 kilometre is 0.013° wide at 10° and 0.016° at 60° — narrow, and not zero.

It is drawn by operation-figure with show: "swap-band" — 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.

53° north, 2° west: four places one pair of numbers could be. The same two numbers, read as three datums and as the pair in the other order. The centre is the WGS84 reading; the marks are where the ground is if the numbers were measured against OSGB36, ED50, NAD27 instead, at 103 m, 138 m, 195 m. Every one of those is a plausible position — near enough to look like a survey difference and far enough to be a different field. Reversing the two numbers instead moves the point 7942 kilometres, and is the mistake nobody worries about because it is usually obvious. What a machine does with it

A coordinate without its system is not a location

The same two numbers name ground 103 metres apart on one datum, 195 on another, and 7,942 kilometres away if the pair is read in the other order. The datum errors are the dangerous ones because they are plausible — and the axis swap, which everyone calls obvious, does nothing at all along a line that crosses three continents.

 A 20° shape across the antimeridian, in the space where the numbers live. Longitude runs across the page from −180° to 180°, which is where the failure is: the shape is one rectangle on the ground and two pieces in the numbering, and every operation that treats longitude as a real number sees the two. The bounding box comes out 359° wide instead of 20°, the planar area comes out 17 times too large because the shoelace encloses the complement, and the midpoint of a segment from one edge to the other lands 20015 kilometres away — the antipode of where it belongs. The true area, from the closed form, is 4,920,667 square kilometres. What a machine does with it

The antimeridian is a cut in the numbers

A twenty-degree box across 180° has a bounding box of 359.4°, a planar area seventeen times too large, and a midpoint 20,015 kilometres from where it belongs — which is the antipode, exactly. Moving the cut moves the failure and never removes it, because a circle cannot be numbered by an interval.

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