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

Two readings of the same triangle, and the band between them

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

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.

It is drawn by operation-figure with show: "inside-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.

3 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.

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.

 The picture is kept, at four tolerances. One closed curve of 3001 vertices, simplified at four tolerances. Douglas–Peucker's promise holds in every panel: no discarded vertex is further than ε from the line drawn in its place, measured at 0.1158 against 0.128 in the last. The picture survives. The enclosed area does not: it falls by 5.43 per cent, and it falls rather than wandering, because cutting a corner takes area off and never puts it back. What a machine does with it

A tolerance is a promise about the picture

Douglas–Peucker guarantees exactly one thing: no vertex it discarded is further than ε from the line drawn in its place. It says nothing about the enclosed area, nothing about which side of the boundary a point ends up on, and nothing about whether the curve still fails to cross itself — and all three are what the geometry is usually being asked.

 A ring round the pole at 80°, and the two pieces it makes. the boundary of a small polar cap — and of everything else. A closed curve divides a sphere into two pieces and neither of them is the outside: one is 4 thousand square kilometres and the other is 506 thousand, a ratio of 130.6 to one, and the coordinates are the same either way. The two colours are the two pieces, sampled at points rather than shaded, because shading one of them would already be the decision this figure is about. What a machine does with it

A polygon on a sphere has no outside

Seven essays have treated a stored ring as a boundary between inside and outside. A closed curve on a sphere divides it into two pieces and neither of them is the outside, so every polygon in every file depends on a convention that no coordinate carries — and the two conventions in common use disagree by a factor of fourteen on any ring that contains a pole.

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