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

A grid at work, and the chain between a tape and a drawing

Sixteen modes, and almost none of them is a map: reduction-chain is the chain itself, tolerance-table and line-scale the quantities a specification names, false-origin and grid-difference and two-conventions the arithmetic of an origin, grid-design and scale-choice and rule-by-direction the design decisions, zone-seam the one discontinuity, closure-blind the check that cannot see itself, setting-out the chain run backwards, surface-grid the grid scaled to the ground, foot-drift the unit that is not one foot, published-against-observed the published coordinate against the measured one, and parcel-area what all of it does to an area.

Sixteen modes, and almost none of them is a map: reduction-chain is the chain itself, tolerance-table and line-scale the quantities a specification names, false-origin and grid-difference and two-conventions the arithmetic of an origin, grid-design and scale-choice and rule-by-direction the design decisions, zone-seam the one discontinuity, closure-blind the check that cannot see itself, setting-out the chain run backwards, surface-grid the grid scaled to the ground, foot-drift the unit that is not one foot, published-against-observed the published coordinate against the measured one, and parcel-area what all of it does to an area.

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 16.

25 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.

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.

false-origin

A false origin moves every number and no geometry 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.

line-scale

A scale factor is a property of a point, and a distance is not Direction decides which rule is adequate, and length does not. The three classical rules for the scale factor of a line, on two lines of the British National Grid at right angles, with the error expressed as distance rather than as a ratio. Bars are logarithmic across four decades. The endpoint mean is out by 11 mm along the 556 km north–south line and by 97 m along the 362 km east–west one — worse on the shorter line by a factor of 8735. In a transverse Mercator k varies with the easting and hardly at all with the northing, so an east–west line traverses the whole parabola and a north–south line sits at one point on it. Simpson's rule holds both to under two millimetres.

rule-by-direction

Direction decides which rule is adequate, and length does not The scale factor was chosen, and the choice is a V. The worst departure of the grid scale factor from unity across a region 9° wide and 9° tall, for every scale factor between 0.998108 and 1.0002, on a central meridian of 2°W. A tangent projection touches at one meridian and is too big everywhere else, at 1987 ppm. Scaling the whole grid down slides that interval until it straddles unity, and the minimum sits at 1/√k_max = 0.999008102, where the worst departure is 993 ppm — a factor of 2.00, which is the most this construction can buy and is reached exactly. The published value marked beside it was chosen the same way, for this region, in the 1930s.

scale-choice

The scale factor was chosen, and the choice is a V What a grid of one's own is worth. For every candidate central meridian, the best worst-case distortion achievable over the region once the scale factor has also been optimised — so each point on the curve is already the bottom of its own V. The minimum is at 2.9°W with k₀ = 0.9993009, giving 700 ppm. The dashed line is UTM zone 30, whose meridian and scale factor were chosen for no region in particular, at 1059 ppm. The national grid is better by a factor of 1.51 — which is the whole of the answer to why a country publishes a grid rather than using the zones, and it is a smaller factor than the argument is usually made to sound.

grid-design

What a grid of one's own is worth Both zones are right, and one of each is not. A 20 km baseline straddling the boundary between UTM zones 31 and 32 at 52°N, computed from grid coordinates three ways, on logarithmic bars. Taking both ends in zone 31 and taking both in zone 32 give answers 0 nanometres apart — the resolution a double has left after carrying a six-figure easting rather than a disagreement — so a job may use either and the overlap belt every zone publishes exists to let it. Taking each end from the zone it nominally belongs to gives 392 km, because the two eastings are measured from meridians six degrees apart and subtracting them measures nothing at all. The same ground point is 706 km east in one zone and 294 km east in the other, and both coordinates are correct.

zone-seam

Both zones are right, and one of each is not 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.

foot-drift

Two feet, and the one that is not a rounding 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.

surface-grid

A grid scaled to the ground, and where it stops being one Four steps between a tape and a drawing. A slope distance of 76.895 km measured at 3.2° on ground 220 m above sea level, reduced to the British National Grid, with every step drawn on a logarithmic scale in millimetres. The slope reduction is the largest by a wide margin at 119.90 m and is the one everybody applies. The grid reduction is 30.23 m. The fourth bar is not a step at all: it is what using the levelled height where the height above the ellipsoid is wanted costs, with a geoid separation of 48.5 m — 583 mm, which is 1.9% of the grid reduction it sits beside and 29 times the tolerance the job closes to. Reading a correction's importance off its share of the chain is how it gets dropped.

reduction-chain

Four steps between a tape and a drawing 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.

closure-blind

A closed traverse cannot see a scale error 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.

two-conventions

Two conventions, and neither of them finds the blunder The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

setting-out

The order the corrections go in, and what it costs 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.

grid-difference

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

tolerance-table

Which corrections a 10 mm job may leave out 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.

published-against-observed

A published coordinate is a definition being quoted 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.

parcel-area

A 1-hectare parcel across British National Grid, at 52°N

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.

 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.

 The scale factor was chosen, and the choice is a V. The worst departure of the grid scale factor from unity across a region 9° wide and 9° tall, for every scale factor between 0.998108 and 1.0002, on a central meridian of 2°W. A tangent projection touches at one meridian and is too big everywhere else, at 1987 ppm. Scaling the whole grid down slides that interval until it straddles unity, and the minimum sits at 1/√k_max = 0.999008102, where the worst departure is 993 ppm — a factor of 2.00, which is the most this construction can buy and is reached exactly. The published value marked beside it was chosen the same way, for this region, in the 1930s. Grids, and what a survey does

The scale factor was chosen

A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.

 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.

 A feature across nine tiles at zoom 4, and the nine labels it gets. A rectangle from -10° to 30° east and 36° to 62° north, drawn in the projected plane with the tile boundaries over it. A renderer that draws each tile from the geometry inside that tile puts the label at the centroid of the piece, marked hollow; the whole feature's centroid is the filled mark. The furthest piece's label is 1863 kilometres from it. The pieces' areas sum to the whole to 2.2e-16 of a relative part, so the clipping is exact and the displacement is the operation rather than an error in it. What a machine does with it

A tile is drawn without its neighbours

Independence is what makes the scheme scale: one request touches one square of the world. It also means a feature crossing nine tiles is nine features, each labelled at its own centroid — up to 1,863 kilometres from the whole one — and the displacement is bounded by the size of the feature rather than the size of a tile, so it grows as the reader zooms out.

 What a grid of one's own is worth. For every candidate central meridian, the best worst-case distortion achievable over the region once the scale factor has also been optimised — so each point on the curve is already the bottom of its own V. The minimum is at 2.9°W with k₀ = 0.9993009, giving 700 ppm. The dashed line is UTM zone 30, whose meridian and scale factor were chosen for no region in particular, at 1059 ppm. The national grid is better by a factor of 1.51 — which is the whole of the answer to why a country publishes a grid rather than using the zones, and it is a smaller factor than the argument is usually made to sound. Grids, and what a survey does

Designing a grid for one region

A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.

 Both zones are right, and one of each is not. A 20 km baseline straddling the boundary between UTM zones 31 and 32 at 52°N, computed from grid coordinates three ways, on logarithmic bars. Taking both ends in zone 31 and taking both in zone 32 give answers 0 nanometres apart — the resolution a double has left after carrying a six-figure easting rather than a disagreement — so a job may use either and the overlap belt every zone publishes exists to let it. Taking each end from the zone it nominally belongs to gives 392 km, because the two eastings are measured from meridians six degrees apart and subtracting them measures nothing at all. The same ground point is 706 km east in one zone and 294 km east in the other, and both coordinates are correct. Grids, and what a survey does

Where two zones meet

A twenty-kilometre baseline across a zone boundary is computed correctly in either zone — the two answers differ by nanometres — and taking one end from each returns 392 kilometres. Both zones are right and the seam between them is not a small error but a category one.

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