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

The scale factor was chosen, and the choice is a V

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

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.

It is drawn by survey-figure with show: "scale-choice" — 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.

4 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.

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.

 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.

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