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

Densification — where it appears

Inserting vertices along an edge so that a stored straight line follows the ground closely enough for a stated tolerance. The tolerance decides the spacing, and the spacing that satisfies a stated ground tolerance depends on the projection the straight line was claimed in.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside 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.

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.

applied · Dataset 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.

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.

applied · Dataset Densifying helps one and hurts the other. The standard deviation of the shoelace area, and the bias of the perimeter, against the number of vertices on the same 100-metre circle with the same 5-centimetre noise on each. The area's spread falls from 9.9 m² to 2.0 — the closed form says it goes as the square root of the vertex count's reciprocal, because a vertex's influence on the area is the vector between its two neighbours and densifying shortens it. The perimeter's bias rises from 0.07 mm to 261, a factor of 3697, because every leg contributes its own σ²/d and shorter legs contribute more.

The area is unbiased and the perimeter is not

A boundary measured from noisy vertices comes out long, always, by σ²/d on every leg. The area enclosed by the same vertices comes out exactly right, because a shoelace is bilinear and the cross terms vanish. So densifying a boundary makes its area five times more precise and its perimeter three thousand times more wrong, and every compactness score computed from it falls short.

distortion · Precision

Named alongside it

The objects these essays reach for when they reach for this one.

ToleranceVerificationGeodesicGreat circleQuadratic lawAggregationAnisotropyBiasBoundaryClosed formConservationConvention

All concepts