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

What a grid of one's own is worth

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

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.

It is drawn by survey-figure with show: "grid-design" — 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.

7 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 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.

 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.

 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.

 The scale spread of every grid Britain could have adopted. A grid is conformal by requirement, so its angular deformation is zero everywhere and the whole design problem is the spread of its one remaining number: the largest scale factor over the region divided by the smallest, in parts per million. The grid's own scale factor does not enter — multiplying every scale by a constant leaves the ratio alone, which is why it is chosen last. The bottom bar is Chebyshev's optimum, the conformal map of this region whose scale is constant on its boundary, which no map of any family can beat; the adopted grid sits 1.98 times above it. Measured over a stated box rather than a coastline, because a coastline would put the vendor's generalisation into the answer. Grids, and what a survey does

The best grid a country could have had

A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.

 The bound was spherical, and the country is not. Three numbers per region, all in parts per million of scale spread. The first is the Chebyshev bound computed on the sphere. The second is that same optimal map used on the ellipsoid, which is what adopting it would actually deliver. The third is the bound with the ellipsoid-to-sphere factor put into the boundary condition, which is the real optimum. The penalty for using the spherical answer reaches 1.22 times — while a named candidate barely moves, because an optimal map has cancelled its own variation and has nothing left to hide a new one in. Grids, and what a survey does

The bound on the body the country is on

The best conformal grid a country could have had was computed against a bound solved on a sphere, with a note saying the flattening was second order and unquantified. It is second order for a named projection — every family's best candidate moves by under 0.7 per cent — and it is 22 per cent for the bound, because an optimal map has already cancelled its own variation and has nothing left to hide a new one in.

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