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

What 0.9996 buys, across one zone at 0°

Drawn here at the parameters it defaults to, with every essay that calls it.
What 0.9996 buys, across one zone at 0°. The point scale factor from the central meridian to the zone edge, measured from the projection's own derivatives. Without the constant the map is exact in the middle and 1382 parts per million too large at the edge. With it the map is 400 parts per million too small in the middle, reaches true scale at 1.75°, and is 981 parts per million out at the edge — a smaller worst case bought by being wrong everywhere.

The point scale factor from the central meridian to the zone edge, measured from the projection's own derivatives. Without the constant the map is exact in the middle and 1382 parts per million too large at the edge. With it the map is 400 parts per million too small in the middle, reaches true scale at 1.75°, and is 981 parts per million out at the edge — a smaller worst case bought by being wrong everywhere.

It is drawn by ellipsoid-figure with show: "grid-scale" — one member of a family of 12 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.

12 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.

 A scale factor is a property of a point, and a distance is not. The grid scale factor sampled along a 362 km line on the British National Grid at a bearing of 90°, with the three rules in use drawn as the constants they replace it by. In a transverse Mercator k depends on the easting almost alone and goes as its square, so along an east–west line the curve is very nearly a parabola. The endpoint mean is the chord across it and lands 268.6 ppm high, which is 97.3 m over this line. Simpson's rule integrates a parabola exactly and lands 0.004 ppm out — 1.3 mm. Neither number is about the line's length. Grids, and what a survey does

The scale factor of a line

A scale factor is a property of a point and a distance is not measured at a point. The three rules in use for averaging one along a line differ by a hundred metres on a 362-kilometre baseline, and which of them is adequate is decided by the line's direction rather than its length.

 Transverse Mercator. The graticule of the Transverse Mercator projection at 30° of longitude and 15° of latitude. the only projection on which a constant compass bearing is a straight line. It is conformal. The families

Transverse Mercator and the series that computes it

The projection most of the world's survey data lives in has no closed form. What every national grid actually computes is a truncated power series in the flattening, and how far it can be trusted is an engineering parameter rather than a property of the projection.

 The sixty zones, each six degrees wide. Every zone is a separate transverse Mercator projection about its own central meridian, so the world is covered by sixty maps rather than one. Zone 31 is picked out, running from 0° to 6° with its axis on 3°. Coordinates do not carry across a zone boundary — a point on either side of one has two entirely different eastings, and nothing in the numbers says which zone they belong to. The families

UTM and the zone system

Sixty separate maps of the world, each six degrees wide, each with a scale factor of 0.9996 chosen so the projection is wrong everywhere and less wrong at the edges. Every constant in the definition is a measured trade rather than a convention.

 What changing the datum alone does to a coordinate. The distance on the ground between a point as read on its national datum and the same numbers read on WGS84, computed through the published seven-parameter transformation. The shifts run from 49 to 166 metres. For comparison, the scale error a UTM zone introduces at its edge is under a metre per kilometre — so the datum, which is usually left unstated, dominates the projection, which is usually argued about. What is taught wrongly

Datum shifts dwarf projection errors

The projection argument is conducted in parts per million and the datum question is answered in hundreds of metres. A coordinate whose datum is unstated is out by more than any projection choice could ever put it, and almost nobody checks.

 A grid scaled to the ground, and where it stops being one. A surface coordinate system on the British National Grid: the grid multiplied by 1.0004044, the reciprocal of the combined factor at a project origin 120 m above the ellipsoid, so that a grid distance equals a ground distance there. It is exact at the origin by construction and wrong everywhere else, and the two directions are not comparable — the axis is logarithmic across five decades. 60 km due east the set-out distance is out by 5559 mm, and the same distance due north by 20.5 mm. The scale factor of a transverse Mercator depends on the easting and almost not at all on the northing, so a system sold as good "within a few kilometres" has a useful area that is a strip rather than a circle. Grids, and what a survey does

A grid scaled to the ground is not a map

Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.

 Grid north against true north at 52°, across a 6° zone. seven meridians drawn in the grid's own coordinates. The grid's north is straight up the page everywhere by construction; the meridians are not, and the angle between them is the convergence. It is exactly zero on the central meridian, reaches 2.36° — 142 minutes of arc — at 3° out, and changes sign across the middle. The manual's γ = Δλ sin φ is 3.0″ short of it at the zone edge. The families

Grid north is not north

A grid has one north, parallel everywhere on the sheet by construction. The Earth has a different one at every point. The angle between them reaches two degrees at the edge of a UTM zone, and a straight line on the grid is not a straight line on the ground either.

 The two reductions, at a grid factor of 1.0004. A measurement made on the ground has to be brought to the ellipsoid and then to the grid, and the two corrections have opposite signs. The elevation factor is R/(R+h) and always shrinks; the grid factor here is 1.0004, which stretches. Their product is the only number a surveyor can use. They cancel exactly at 2551 metres. At 1500 metres the combined factor is 165 parts per million, which is 1.65 metres on a 10 kilometre baseline. What the numbers refer to

The ground is not the grid

A tape measure on a hillside has to be brought down to the ellipsoid and then out onto the map, and the two corrections have opposite signs. On a grid whose scale factor exceeds one there is exactly one elevation where they cancel — 2,551 metres, for a factor of 1.0004.

 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.

 Which corrections a 10 mm job may leave out. Each correction inverted: the line length — or, for the last row, the patch radius — at which that correction alone reaches 10 mm, for a job 150 m above the ellipsoid on 2° of ground, 0.0° from the British National Grid's central meridian. The tightest is the slope reduction at 16 m. Three of the four rows are linear in the tolerance, so this ranking is the same at a millimetre and at a decimetre; what does change it is the site — move the job up a mountain or onto steeper ground and the order is different, which is why a specification's list is not transferable. Grids, and what a survey does

The tolerance decides the model

Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.

 The boundary of the region the series is the map in. The curve where the transverse coordinate reaches 2.918, which is where the terms stop shrinking. It crosses the equator 83.81° from the central meridian and closes towards the poles, because the same longitude is a smaller transverse coordinate at a higher latitude — the boundary is a curve rather than a meridian. The narrow band beside it is a 3° zone, the width national grids actually use, drawn to the same scale: the practical world sits in about a fiftieth of what the series can reach. Drawn in Mollweide. The families

Where the series stops being the map

The transverse Mercator has no closed form on an ellipsoid, so every national grid computes a truncated series. Asking how many terms it needs has an answer everywhere; asking whether more terms always help has an answer only within 83.81° of the central meridian, and the boundary is computed rather than assumed.

 The widest zone a tolerance of 690 parts per million allows. At each latitude, the half-width at which a transverse Mercator grid with its scale factor rebalanced for that width reaches 690 ppm at its worst point. That tolerance is the one UTM actually meets at the equator, so the curve passes through UTM's own 3° there — and rises to 37.0° at 85° north, because a degree of longitude covers cos φ of the ground and the scale error goes as the square of the ground width. Six degrees is the answer at one latitude. Grids, and what a survey does

Sixty zones was a decision about one latitude

Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.

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