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

Only one of a grid's five parameters changes the map

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

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.

It is drawn by survey-figure with show: "grid-difference" — one member of a family of 16 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.

9 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.

A false origin moves every number and no geometry. British National Grid, drawn twice at the same scale. On the left the coordinates are measured from the projection's own origin, where the central meridian meets the true origin's parallel, and 52% of the country takes a negative easting or northing — the worst reaching -323 km. On the right the authority's published false origin of 400 km east and -100 km north has been applied, and none of them does. Every distance computed from the two sets of numbers agrees to the last bit a double has left after carrying six figures, and every bearing agrees exactly; the only thing that changed is that no coordinate carries a sign. The margins say the origin was fitted to the land rather than placed arbitrarily below it: 77 km spare in the west and 50 m in the south, on a grid 988 km tall. Grids, and what a survey does

A grid has an origin that is not there

Every published national grid measures from a point in the sea. The British one sits 400 kilometres west and 100 kilometres south of where the projection's own zero is, and moving it changes every coordinate in the country and not one distance or bearing.

 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.

 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.

 53° north, 2° west: four places one pair of numbers could be. The same two numbers, read as three datums and as the pair in the other order. The centre is the WGS84 reading; the marks are where the ground is if the numbers were measured against OSGB36, ED50, NAD27 instead, at 103 m, 138 m, 195 m. Every one of those is a plausible position — near enough to look like a survey difference and far enough to be a different field. Reversing the two numbers instead moves the point 7942 kilometres, and is the mistake nobody worries about because it is usually obvious. What a machine does with it

A coordinate without its system is not a location

The same two numbers name ground 103 metres apart on one datum, 195 on another, and 7,942 kilometres away if the pair is read in the other order. The datum errors are the dangerous ones because they are plausible — and the axis swap, which everyone calls obvious, does nothing at all along a line that crosses three continents.

 A closed traverse cannot see a scale error. The same closed figure with two different errors in it. On the left every leg is 400 parts per million too long, which is roughly what forgetting the grid scale factor costs — and the figure closes to 2.3e-13 m, which is the last bit of a double rather than a measurement. Scaling every leg of a closed figure by the same factor produces a similar figure and a similar figure is still closed, so the check every specification leans on is blind to it. Every dimension in that figure is wrong: the perimeter is out by 1.17 m. On the right one angle is 20 seconds out — a far smaller disturbance in its own units — and the figure fails to close by 0.062 m. Closure tests the shape and says nothing about the size. Grids, and what a survey does

A traverse must close

The check every survey specification leans on cannot see the error this whole field is about. Scale every leg of a closed figure by four hundred parts per million and it closes to the last bit of a double, while every dimension in it is wrong.

 Two conventions, and neither of them finds the blunder. A blunder of 350 mm added to leg 3 of a closed traverse, producing a misclosure of 350 mm — one part in 8344, which most specifications would accept. Bowditch's rule shares the misclosure out in proportion to leg length; the Transit rule shares it in proportion to each leg's component along the axis being corrected. Both close the figure exactly, so both are valid; they disagree with each other by up to 14 mm; and neither puts more than 104 mm of correction on the leg that is actually wrong. A rule for distributing a misclosure is a convention for producing consistent numbers, and it is not an instrument for finding errors. Grids, and what a survey does

The two ways to spread a misclosure

Bowditch's rule and the Transit rule take the same closed figure and the same misclosure and disagree about which legs were wrong. Both close it exactly, neither puts the correction on the leg that actually carries the blunder, and no measurement can settle which is right.

 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

Two grids over the same ground

One point in the English Midlands is 406,788 east on the British grid and 167,478 east on UTM. Separating the difference properly leaves a surprise — only one of a grid's five parameters changes the map, and it is not any of the four that dominate the number.

 A published coordinate is a definition being quoted. Re-observing a control point perfectly, with an instrument good to 20 mm, and comparing the answer to what is published for it. The published coordinate was fixed on a datum realisation that has since been superseded, which accounts for 106 m, and the ground it marks has moved 750 mm in 30 years at 25 mm a year. Both numbers are larger than the observation and neither is an error in it: subtracting a published coordinate from an observed one measures the interval between two definitions. Grids, and what a survey does

A published coordinate is a result

Re-observe a control mark perfectly, with an instrument good to twenty millimetres, and the answer disagrees with the published value by a hundred metres. Neither number is wrong. The difference measures the interval between two definitions.

 A 1-hectare parcel across British National Grid, at 52°N. The departure of a plan's dimensions from the ground's, across the width of the zone, as a length and as an area. The area curve is the length curve doubled — an areal scale factor is the square of a linear one, and squaring a small departure doubles it — so a parcel whose sides are each 399 parts per million short on the central meridian is 797 short in area. On a 1-hectare parcel that is 8.0 square metres at the central meridian and 8.0 at 0.00° out. Grids, and what a survey does

An area on the grid is not an area on the ground

A grid's scale factor is a property of lengths and what a surveyor sells is an area. Squaring a departure doubles it, so a hectare drawn on the British grid at its central meridian has 10,008 square metres of ground under it — and correcting an area with the line factor instead of its square leaves half the error behind.

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