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

A datum, and the disagreement between two of them

datum-spread and parameter-bars are the shift itself, fit-residuals and local-fit what a fit leaves behind, shortcut-error what skipping the fit costs, grid-spacing the resolution a shift grid is published at, plate-map and epoch-drift the part that moves with time, degree-length and arc-inversion the historical measurement that produced the figures, and parameter-freedom the directions a seven-parameter fit cannot pin down.

datum-spread and parameter-bars are the shift itself, fit-residuals and local-fit what a fit leaves behind, shortcut-error what skipping the fit costs, grid-spacing the resolution a shift grid is published at, plate-map and epoch-drift the part that moves with time, degree-length and arc-inversion the historical measurement that produced the figures, and parameter-freedom the directions a seven-parameter fit cannot pin down.

Every one of these is the same generator answering a different question, which is why they share a file, a set of colour roles and a set of assertions. Changing it changes all 11.

30 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.

What it draws

One picture per question the family answers. Each has a page of its own.

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.

datum-spread

The same coordinate on four datums OSGB36 to WGS84, one parameter at a time. Each bar is how far the mark at 2.0° west, 54.5° north moves under one of the seven parameters with the other six set to zero. seven of the seven are not zero for OSGB36. The units hide the comparison: one arcsecond of rotation moves this point 30.8 metres and one part per million of scale moves it 6.36 metres, so OSGB36's scale term contributes 130 metres — more than one of its three translations.

parameter-bars

OSGB36 to WGS84, one parameter at a time What 4 parts per million of network strain leaves behind. Each arrow is where the best-fitting seven-parameter transformation leaves a marker, over 36 markers laid out as a grid across OSGB36's ground. The RMS residual is 1.64 metres and the worst is 3.00 metres. Seven parameters span the constant and linear parts of a displacement field; this one is quadratic, so no choice of the seven can reach it. The arrows are drawn 27× life size.

fit-residuals

What 4 parts per million of network strain leaves behind Molodensky against the exact route, NAD27. The distance on the ground between where each shortcut puts the transformed point and where the exact Cartesian route puts it, at 0 metres of ellipsoidal height, on a logarithmic scale. The full formulae stay within 0.8 centimetres across every latitude drawn; the abridged form, which replaces two ellipsoid-difference coefficients with one combined term, is worst at 40 centimetres — a factor of 48. The abridged curve dips near 45°, where the combined coefficient happens to equal the pair it replaces.

shortcut-error

Molodensky against the exact route, NAD27 What grid spacing a tolerance buys. The worst bilinear interpolation error across OSGB36's ground, against the spacing of the table it is interpolated from, both axes logarithmic. The smooth shift falls by exactly four for every halving, which is what second-order interpolation does. Adding a 30 centimetre ripple 2 degrees across — three parts in a thousand of the 99 metre shift it rides on — moves the spacing needed for 20 millimetres from 1° to 0.125°, which is 64 times as many nodes.

grid-spacing

What grid spacing a tolerance buys Where the ground is going, six plates. The velocity of the ground under a coordinate, computed as the cross product of each plate's rotation vector with the position, and drawn along the band where that rotation is fastest rather than over an outline this site does not hold. The quickest arrow is 75 millimetres a year on the Pacific plate, which is 1.89 metres in twenty-five years. drawn in Robinson — the arrows' directions are the projection's as much as the plates'.

plate-map

Where the ground is going, six plates A coordinate with no date on it. Ground displacement against elapsed time for five places, each on its own plate, computed from the plate's rotation vector. Honolulu moves 71 millimetres a year and reaches 2.85 metres in 40. The dashed line is 10 centimetres, which is the tolerance an ordinary boundary survey works to: every one of these crosses it, and three of them cross it within five years.

epoch-drift

A coordinate with no date on it One degree of latitude, on three ellipsoids. The ground length of one degree of latitude, integrated from the meridian radius of curvature. On WGS84 it runs from 110574 metres at the equator to 111694 at the pole — a rise of 1120 metres, which is the entire signal that separates a flattened Earth from a spherical one. The degree is longer where the surface is flatter, which is at the pole, and the ordering catches out anybody reasoning from the outline of the meridian ellipse.

degree-length

One degree of latitude, on three ellipsoids How badly two arcs determine the flattening. The reciprocal flattening recovered by inverting two measured degree lengths, at 1.5° and 66.33°, against an error introduced into the equatorial one. The exact pair returns 298.26. Ten metres of error — 90 parts per million of a 110-kilometre arc — returns 301.5, and the relative error in the flattening is 118 times the relative error in the arc. At 1000 metres the inversion returns a negative flattening: an Earth longer through the poles than across the equator, which is the answer the Paris Observatory defended for a generation.

arc-inversion

How badly two arcs determine the flattening Fitting a sphere to 10° of latitude at 54.5°. The Earth's Gaussian radius of curvature across the band, with two candidate spheres drawn against it. The global mean radius is 6371.0 kilometres and sits 2203 parts per million away from the ground here — 2.2 metres in every kilometre measured. The best local radius is 6385.0 kilometres and has no bias at all by construction, leaving 325 parts per million of spread that no sphere can remove, because the curvature varies across the band and a sphere's does not. The gain is a factor of 6.9.

local-fit

Fitting a sphere to 10° of latitude at 54.5° Two parameter sets 100 metres apart, over the region they were fitted to. Each marker is drawn at a size proportional to how far the two transformations put it apart. The second set differs from the first by 100 metres of translation along the direction this network can least see, with the rotations and the scale re-fitted to absorb it — which is what a second agency's adjustment does when it chooses a different constraint. The worst disagreement anywhere in the region is 5.64 metres and the mean is 3.61. Applied at south-eastern Australia the same two sets differ by 193 metres, because the rotation that absorbed the translation here is a rotation of the whole Earth.

parameter-freedom

Two parameter sets 100 metres apart, over the region they were fitted to

Where it is called

Changing this changes every one of these figures.

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 the numbers refer to

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.

 OSGB36 to WGS84, one parameter at a time. Each bar is how far the mark at 2.0° west, 54.5° north moves under one of the seven parameters with the other six set to zero. seven of the seven are not zero for OSGB36. The units hide the comparison: one arcsecond of rotation moves this point 30.8 metres and one part per million of scale moves it 6.36 metres, so OSGB36's scale term contributes 130 metres — more than one of its three translations. What the numbers refer to

The seven parameters, and what each one does

A datum transformation is published as three metres, three arcseconds and one part per million, in three different units — which hides the fact that all seven are the same size of thing. Converted to ground displacement, the scale term beats two of the translations.

 Only one of a grid's five parameters changes the map. The same ground point written on the British National Grid and on UTM zone 31N, with the difference between the two coordinate pairs taken apart. The largest term by four orders of magnitude is where the two grids put zero — 5544 km, being a different central meridian, a different true origin and different false constants, all of which move every coordinate and no distance. The datum, which is the term everybody names, moves the ground 106 m. And the whole geometric difference between two transverse Mercators is their scale factors — 0.9996012717 against 0.9996 — which at this point is 50 cm. Four of a grid's five parameters are bookkeeping; the fifth is the map. Grids, and what a survey does

What a grid is made of

A projected coordinate declares five things — a datum, a projection, its parameters, a unit and an axis order — and four of them are invisible in the numbers. Every one has a documented case of being guessed wrong, and the cheapest of those cost a hundred metres.

 Fitting a sphere to 10° of latitude at 54.5°. The Earth's Gaussian radius of curvature across the band, with two candidate spheres drawn against it. The global mean radius is 6371.0 kilometres and sits 2203 parts per million away from the ground here — 2.2 metres in every kilometre measured. The best local radius is 6385.0 kilometres and has no bias at all by construction, leaving 325 parts per million of spread that no sphere can remove, because the curvature varies across the band and a sphere's does not. The gain is a factor of 6.9. What the numbers refer to

A datum is fitted to a region

Forty countries adopted forty ellipsoids, and the usual explanation is that measurement was poor. It was not — a figure fitted to one country beats the global one over that country by a factor of seven, and what it removes is a systematic bias of two metres in every kilometre.

 Why there are two latitudes, at 45°. A meridian section of an ellipsoid with the flattening exaggerated 12× — at the true value of 1/298 this outline would be indistinguishable from a circle. Geodetic latitude is the angle the surface normal makes with the equatorial plane; geocentric latitude is the angle the radius makes with it. The two lines meet the plane at different points and the angles differ by 11.55 arcminutes on the real WGS84 ellipsoid, which is about 21 km of ground distance. What is taught wrongly

The Earth is a sphere, and when it is not

Every essay before this one treated the Earth as a ball, and said so. The flattening is one part in three hundred, which is nothing for a distance, everything for a latitude, and exactly enough to make the most-used projection in the world fail the property in its own name.

 What 4 parts per million of network strain leaves behind. Each arrow is where the best-fitting seven-parameter transformation leaves a marker, over 36 markers laid out as a grid across OSGB36's ground. The RMS residual is 1.64 metres and the worst is 3.00 metres. Seven parameters span the constant and linear parts of a displacement field; this one is quadratic, so no choice of the seven can reach it. The arrows are drawn 27× life size. What the numbers refer to

Where a fit leaves residuals

Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.

 Molodensky against the exact route, NAD27. The distance on the ground between where each shortcut puts the transformed point and where the exact Cartesian route puts it, at 0 metres of ellipsoidal height, on a logarithmic scale. The full formulae stay within 0.8 centimetres across every latitude drawn; the abridged form, which replaces two ellipsoid-difference coefficients with one combined term, is worst at 40 centimetres — a factor of 48. The abridged curve dips near 45°, where the combined coefficient happens to equal the pair it replaces. What the numbers refer to

Molodensky's shortcut

The exact way to shift a datum needs an iteration nobody wanted to run on a 1950s machine, so a direct formula was derived instead. It lands within centimetres — and the abridged version everybody quotes is a different formula, worse by a factor of fifty.

 Two feet, and the one that is not a rounding. The US survey foot is 1200/3937 m and the international foot is exactly 0.3048 m, so the first is longer by 2.0000 parts per million. Read a coordinate in the wrong one and the error is proportional to the coordinate, not to any distance in the job: at 2.5 million feet from the false origin it reaches 1.52 m, and it crosses a survey's own closing tolerance of twenty millimetres at 33 thousand feet. Across a 300-foot site the same mistake changes every dimension by 0.18 mm — below every tolerance there is, which is why a plan in the wrong foot passes every check and is in the wrong place. Grids, and what a survey does

The units are part of the coordinate

Two feet were in use in the United States until 2022, differing by exactly two parts per million. A plan drawn in the wrong one has every dimension right to a fifth of a millimetre and sits three-quarters of a metre from where it belongs, which is why it passes every check anybody runs.

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