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

Coordinate semantics — where it appears

What a coordinate's numbers actually refer to — which body, which realisation of which datum, at which epoch, in which order and which units. It is the part of a position that software routinely discards and a reader routinely assumes.

Named by 8 essays across 4 fields — each of them below, with the objects they name alongside it.

The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres. The numbers are identical; only what they refer to differs.

What a coordinate refers to

A latitude and a longitude are not a place. They are a claim about a model, and the model has a shape, a position, a set of marked stones and a date — change any one of the four and the ground under the numbers moves by hundreds of metres.

datums · Datum Five latitudes that are not the latitude, on WGS84. Each curve is the amount by which one auxiliary latitude falls below the geodetic latitude a coordinate actually means, in arcminutes. All five vanish at the equator and at the poles and peak near 45°, where the geocentric latitude is 11.55 arcminutes below the geodetic one — about 21.4 km on the ground. The curves never cross, which is forced by the algebra rather than by this ellipsoid's particular flattening.

Geodetic against geocentric latitude

Two angles, both called latitude, differing by eleven and a half arcminutes at their worst. One is what every coordinate means and the other is what every spherical formula assumes, and the gap between them is twenty-one kilometres on the ground.

wrong · Ellipsoid 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 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.

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.

applied · Dataset What simplifying a boundary does to the number stored beside it. One region, simplified at five tolerances, with the error in the two quantities a consumer computes from the pair. If the density was stored, the total it implies moves by exactly the area's error — -1.55 per cent at the loosest tolerance. If the total was stored, the density it implies moves the other way by the same amount. Nothing in the file says which of the two was measured and which is being derived, and the simplification is normally done by a tool that never opens the attribute table.

The attribute is a claim about the geometry

Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.

applied · Dataset A section through a body with a neck, and the rays that leave it twice. An equatorial section of a stated contact binary — two lobes of radius 1 centred at ±1.5, joined by a neck of radius 0.35, with the origin in the neck. The lines are rays from the origin and the marks are where each one crosses the surface. two of the 25 drawn cross more than once, so along those directions there is no such thing as "the" radius, and a longitude and a latitude do not name a place.

A ray from the centre hits the surface twice

Eleven rungs map bodies that are lumpy, triaxial and turning at a drifting rate, and every one assumes the surface is star-shaped about the centre — which is what makes a longitude and a latitude a coordinate at all. A contact binary is not: on a stated body with a neck a third of a lobe wide, 10.9 per cent of the sky has no single radius, and the shape model everybody publishes fills the neck in and adds 1.67 per cent of the volume.

datums · Bodies Millimetres of horizontal move per kilometre of assumed height. How far the transformed latitude and longitude move when the height fed into a datum transformation changes by one kilometre, for four published transformations at five places. It runs from 7.7 to 121 millimetres per kilometre. A latitude and a longitude do not carry a height, so a two-dimensional coordinate cannot be transformed until somebody supplies one that is not in it.

The height a coordinate does not carry

Ten rungs treat a datum shift as a map from one pair of angles to another. It is not one: the transformation runs through Cartesian coordinates, so the answer depends on the height — by 7.7 to 121 millimetres per kilometre depending on the datum and the place, which puts Lhasa 432 millimetres from where a height of zero would have said.

datums · Datum Two places, one written coordinate. The lattice a coordinate written to 5 decimal places lives on, at 45° of latitude, where a cell is 1.11 metres north to south. Two real places a fifth of a cell apart are drawn as open marks and the single value they both round to as a filled one. The distance between them, computed from what was written, is exactly zero — an error a noise model cannot produce, since independent noise never puts two different points in the same place.

Rounding is not noise

Eight rungs treat a coordinate's error as noise that averages down. A published coordinate has a second error that does not: it is deterministic, it is shared between every point in the same cell, and at five centimetres apart ninety-three per cent of pairs come out as the same place — which no amount of independent noise can produce.

distortion · Precision

Named alongside it

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

ConventionVerificationDatumEllipsoidFlatteningToleranceAreaBoundaryClosed formGeodetic datumHelmert transformationTopology

All concepts