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

Realisation — where it appears

The set of marked points and published coordinates that turn a datum's definition into something a receiver can be positioned against. It is what a receiver is positioned against, and it carries an epoch, because the marked points move with the plate they are on.

Named by 29 essays across 7 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 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.

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.

datums · Datum 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.

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.

practice · Grid 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.

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.

datums · Datum 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.

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.

datums · Datum 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.

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.

practice · Grid 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.

When a formula is not enough

National mapping agencies distribute datum shifts as tables of numbers on a grid rather than as parameters. What decides the spacing that table needs is not the size of the shift — a hundred-metre shift tabulates coarsely, and a thirty-centimetre ripple riding on it does not.

datums · Datum 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'.

The epoch is part of the coordinate

A latitude and a longitude with no date on them are incomplete, because the ground they refer to is moving at between ten and seventy millimetres a year. Australia crosses an ordinary survey tolerance in under two years and moved 1.8 metres between two versions of its own datum.

datums · Datum 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.

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.

applied · Dataset Three surfaces, and the two heights between them. The ellipsoid, the geoid and the ground, with the two heights a coordinate can carry. Ellipsoidal height h is what a satellite fix returns and is measured from a surface defined by four constants. Orthometric height H is what a level and a staff measure and is referred to the geoid — the equipotential surface that best fits mean sea level. They differ by the separation N, drawn here as 45 metres because that is a stated input rather than a computed one: a geoid model is a data product with a truncation degree in it, and this site computes rather than downloads. The arithmetic h = H + N is exact whatever N is.

Height above what?

A satellite reports a height above a mathematical surface. A level and a staff report a height above the surface water settles on. The two disagree by tens of metres, both are correct, and only one of them decides which way a pipe drains.

datums · Height Why a levelled height is not a distance. The correction between raw levelling and orthometric height, for lines at 200, 500, 1000, 2000 metres above the geoid running north from 50°. It is not instrument error: level surfaces converge towards the pole, so a run that stays on one of them gains height relative to another. A line 2000 metres up reaches 647 millimetres over 400 kilometres. The dashed line is 10 millimetres, which is about what a first-order levelling network closes to over that distance — so this is not a refinement, it is the larger of the two numbers.

A levelled height is not a distance

Level surfaces converge towards the poles by five metres in a thousand, so a chain of perfectly executed levelling observations does not sum to a height difference. The correction over four hundred kilometres of northing is larger than the network's own closure.

datums · Height A deflection of 10 arcseconds, and what it hides. The ellipsoid normal and the plumb line at one point, with the geoid tilted against the ellipsoid by 10 arcseconds — drawn 3000× steeper than life, because at true scale the two lines are indistinguishable. The relation is exact and linear: an arcsecond of deflection is the geoid rising 4.85 millimetres in a kilometre, so 10 arcseconds is 48.5 millimetres per kilometre. A star sight measures the plumb line's direction, so astronomic latitude differs from geodetic by exactly this angle — 309 metres of ground, at a point where the coordinate itself is correct.

The plumb line is not the normal

A latitude measured from the stars and a latitude that means a position on the ellipsoid are different angles, because a plumb bob hangs along gravity and gravity is not perpendicular to a mathematical surface. Ten arcseconds of difference is 309 metres of ground.

datums · Height 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, and the height changes by 51, 44, 47 metres, which is the component nobody quotes.

The third coordinate moves too

A datum shift is quoted as a horizontal displacement because horizontal is what people look at. The transformation acts on a three-dimensional point, and its vertical component is between a quarter and a half of the horizontal one — 51 metres, for a British coordinate.

datums · Height 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.

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.

practice · Reduction What a solved map is, as a list of numbers. A map with no formula is a list of coefficients, and this is the list. The Chebyshev map of an elongated region, 30° by 10° has its coefficients falling by a factor of 1.5e+13 from the first to the fourteenth, so a table of a dozen numbers carries the whole projection; the conformal cube face's coefficients, marked separately, fall far more slowly because the map has a singularity at each corner. How fast this line falls is exactly how portable the map is — and neither map has a name, an inverse in closed form, or a formula anybody could quote.

A map with no formula

The solved projection has no name, no formula and no closed-form inverse. It is fourteen numbers — and the rate at which those numbers fall away decides whether a map can be shipped at all: geometrically for a smooth region, and like a power for one with corners.

choosing · Condition 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.

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.

practice · Reduction

Two parameter sets, one transformation

Agencies publish seven-parameter datum transformations that differ by hundreds of metres in translation, and the usual reading is that one of them is better. Over the region either was fitted to they are the same transformation: a hundred metres of translation, re-absorbed by the rotations and the scale, moves a British coordinate by 5.6 metres and an Australian one by 193.

datums · Datum

How big a triangle it takes

Gauss proved that a surface-dweller can read the curvature off a triangle's angles. Doing it is another matter: a fifty-kilometre triangle has 5.49 seconds of excess, a one-second theodolite gives that excess a standard deviation of 1.73, and reading K to one per cent needs a side of 281 kilometres. Not one of the great surveys built a triangle within a factor of three of that.

impossibility · Curvature

A body with no sea level

On Earth the zero of height is found rather than chosen — water settles onto the equipotential surface by itself. Nowhere else has one, and the difference is measurable: Clairaut's theorem predicts the Earth's flattening from its own gravity field to twelve metres at the pole and misses Mars's by 2.2 kilometres.

datums · Bodies

A chain of transformations does not close

Two datum transformations applied one after the other are not the sum of their fourteen published numbers. The gap is seven millimetres in Britain, doing the two in the other order moves the answer twenty, and the rotation matrix everybody prints is not a rotation.

datums · Datum

One pair of numbers, a hundred and twenty places

A UTM coordinate is two numbers and a zone. Drop the zone and the numbers are still valid in each of the sixty; drop the hemisphere too and the pair names a hundred and twenty places. They form two rings at one latitude each, spaced exactly six degrees apart, and every one of them has the same grid convergence and the same scale factor — so no further geometric measurement can choose between them.

practice · Grid

The answer is a set

Eight rungs have produced a best fit — one projection, ranked first, with a margin. A best fit without a spread is not a measurement, and the spread is free: a control point has a width, and every candidate whose residual is inside that width has not been ruled out. At one per cent noise a four-degree region admits ten of twenty candidates and a forty-degree one admits exactly one.

wrong · Identify

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

datums · Height

A velocity needs a frame and a strain rate does not

One place on a plate boundary moves at 5.3, 7.0, 10.2 or 54.6 millimetres a year, and towards the south-east or the north-west, depending on which rotation was subtracted first. Every one of those readings reports the same strain rate, to two parts in a hundred thousand million.

datums · Strain

A grid stops fitting the ground it was laid on

A hundred-kilometre baseline across a plate boundary changes by 815 millimetres in fifty years, and the same baseline on the fastest plate in the model changes by nothing at all in the same fifty. The interval before an ordinary first-order specification is broken is six years in one place and never in the other.

datums · Strain

Every country's zero is a different surface

Ten rungs measure a height against a geoid and treat the geoid as one object. No national datum is on it: each is pinned to the mean sea level at one tide gauge, the sea surface stands up to two metres from the geoid, and the eight European zeros measured here spread over 462 millimetres — a step no levelling can remove and nobody's error.

datums · Height

A tolerance in map units is not a tolerance

A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.

applied · Dataset

One number changed and the whole map moved

Raise one bump's weight in a density specification, leave every other number identical, and solve again. London's own value is what changed; London moves 0.054 and Delhi moves 0.226 — four times as far, with its own number untouched. The largest displacement anywhere is thirty degrees from the change, and the antipodal band still moves a fifth of the peak.

distortion · Cartogram

The fourth number the ellipse does not carry

A Jacobian has four independent entries and an indicatrix reports three. The missing one is the rigid rotation in the polar decomposition A = R·S, the indicatrix is exactly S, and R is what turns north into grid north: for the transverse Mercator it agrees with the survey formula for convergence to 5 × 10⁻¹⁰ degrees, from a different library and a different derivative.

distortion · Tissot

Named alongside it

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

ToleranceVerificationDatumConventionNational GridOSGB36Closed formHelmert transformationWGS84GeoidScale factorDegeneracy

All concepts