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

"Within 0.01°" on the ground, at five latitudes

Drawn here at the parameters it defaults to, with every essay that calls it.
"Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west.

The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west.

It is drawn by operation-figure with show: "ground-ellipse" — 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.

6 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.

One pixel at zoom 11: 76.4 m projected, 47.6 m at 51.5°. The grid squares are pixels, at their own size. The open mark is the stored coordinate and the filled one is where it is drawn: rounding moves it 20.2 metres, against a worst case of 33.6 — half a pixel's diagonal, which is the whole of the bound. The second point sits 43 metres away, 0.58 of a pixel east and 0.70 north, so whether the two are drawn as one dot or two is decided by where the tile grid happens to fall: they merge for 12 per cent of the possible offsets, against the 12 per cent the two fractions predict. What a machine does with it

The pixel is a place with a size

Drawing a coordinate rounds it to a pixel, which moves it by up to half a diagonal — 16.8 metres at zoom 12. Whether two points 43 metres apart appear as two dots is not a property of the data at all: they merge for 12 per cent of the positions the tile grid could take, and the closed form predicts 12.4.

 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. What is taught wrongly

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.

 "Within 0.01°" on the ground, at five latitudes. The same condition drawn at five latitudes, all at one scale. A degree of latitude is a fixed distance — 1106 metres here, varying by less than one per cent from equator to pole — while a degree of longitude collapses as cos φ, from 1113 metres to 289 at 75°. So the "circle" is an ellipse of 3.86:1 there, and it encloses 26 per cent of the ground the same condition covers on the equator. Even on the equator it is not round: M is smaller than N by the flattening, so the shape is 6694 parts per million shorter north–south than east–west. What a machine does with it

A degree is not a unit of length

"Within 0.01 degrees" is a condition anybody can write and no instrument can measure. On the ground it is an ellipse — 1,106 metres north–south and 558 east–west at 60° — and even on the equator it is not a circle, because the meridian's radius of curvature is smaller than the parallel's by the flattening.

 5 of 400 nearest-neighbour queries change answer in the plane. 40 sites and 400 queries over -20° to 40° east and 35° to 70° north, each query answered twice — once by geodesic distance and once by straight-line distance in the stored plane. They agree 98.8 per cent of the time, which is why the operation survives, and the 5 that differ are marked. The mechanism is not that the plane is wrong by a lot but that its scale factor varies: over this region it spans 139 per cent, and every disagreement is a contest closer than that — the worst margin measured is 6.4 per cent. The furthest a wrong answer is from the right one is 24 kilometres. What a machine does with it

Nearest is a question about the metric

Asked in the plane the data is stored in, a nearest-neighbour query returns a different site for 5 of 400 queries over Europe on Web Mercator — and 103 of 400 on the plate carrée, with the wrong answer up to 309 kilometres further away. Every disagreement is a contest closer than the region's own scale spread, and over a city there are none.

 five planes a dataset might be stored in, scored on three operations. Each candidate measured over -10° to 30° east and 35° to 60° north: the worst areal error, the worst angular deformation, and the spread of the scale factor, which are what an area query, a shape and a distance respectively depend on. The best plane for areas is Gall–Peters, for shapes Lambert conformal conic, and for distances Lambert conformal conic — three different answers, and no fourth candidate would collapse them, because a projection exact in two of these columns has a = b = 1 everywhere and is the isometry Gauss's theorem forbids. area of a polygon costs 3.06× too large in the wrong plane; drawing a line between two points costs 194 km from the ground it claims. What a machine does with it

The operation decides the coordinate system

Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.

 What the 5th decimal place of a coordinate is worth on the ground. A latitude written to 5 decimal places steps 1.112 m north for one unit in its last digit, at every latitude, because the meridian does not care where it is measured. A longitude written to the same 5 places steps 1.112 m east on the equator and 0.097 m at 85°, because a degree of longitude is a degree of a circle whose radius is R cos φ. The same written precision means two different distances at the same point, and a different pair at every other. Measuring distortion

A coordinate is a number with a width

Every number on this site so far has been exact. A written coordinate is not: five decimal places of a degree is 1.112 metres of latitude everywhere and 1.112 metres of longitude only on the equator, falling to 0.097 at 85°. The same written precision carves the ground into a cell that is square in one place and eleven times longer than it is wide in another.

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