The best grid a country could have had

A country choosing a grid is choosing a conformal map of one region. Conformality is not negotiable — a surveyor needs one scale factor at each point rather than two, so that a distance can be corrected without knowing which way it runs — and once it is imposed, the angular deformation is zero everywhere by construction and there is nothing left to optimise but the spread of the one remaining number.

Designing a grid for one region compares two named options and finds that a country’s own grid beats the international zone that would otherwise carry it by a factor of 1.51. That is a comparison between two candidates. This rung asks the question the comparison implies and does not answer: what is the best that any conformal map could have done, and how far from it is the grid that was adopted?

The quantity a grid is designed against #

The scale spread has one property that makes it the right measure and is worth stating because it is the reason the design problem splits in two. The grid’s own scale factor does not enter it. Multiplying every scale factor by a constant leaves the largest divided by the smallest alone.

That is why k₀ is chosen last and separately, and why the scale factor was chosen can derive Britain’s 0.9996012717 as the reciprocal square root of the worst distortion a tangent projection would have had. The projection and its parameters decide the ratio; k₀ decides where in that ratio unity sits, and the best it can do is put the map exact on two lines and split the error evenly either side.

So the design question is: over all conformal maps of this region, which one has the smallest ratio?

The bound #

Chebyshev answered it in 1856, and this site already carries the machinery: the conformal map of least scale spread over a region is the one whose scale is constant on the region’s boundary. Chebyshev’s criterion states it and verifies it against the one case with a closed form, the spherical cap, where the answer is the stereographic projection; solving for the map instead of choosing it turns it into a least-squares fit that solves for any region at all.

Pointing that solver at a country gives a number no named projection can go below, and the comparison is then a real one rather than a beauty contest between two candidates.

For the wider of the two boxes used here, the bound is 1,066 parts per million. The best named projection reaches 1,442, and the adopted grid 2,107.

That no candidate falls below the dashed line is the check that the bound is a bound. It would be a much weaker figure if the solver had produced a number some ordinary projection could beat, and that outcome is exactly what a mis-scaled or mis-framed fit would produce.

What the one parameter is worth #

A transverse Mercator has one parameter that matters to the spread, and the curve above is the whole of what it is worth. The shape is the practical content. Near the bottom the curve is quadratic and therefore flat — an error of half a degree in the central meridian is invisible — and away from it the cost grows quickly.

That is a common shape in optimisation and it has a specific consequence here: the choice can be made badly and still be nearly right, which is why national grids adopted before anybody could compute this are mostly fine. It also means the adopted grid’s apparent shortfall needs a much more careful reading than a bar chart gives it.

The region is the specification #

Here is the finding, and it is not the one this essay set out to make.

Scored over a box from 8° west to 2° east, Britain’s grid is 46 per cent worse than the best transverse Mercator, whose central meridian would be at 3° west rather than 2°. Scored over a box from 6° west to 2° east, the adopted grid is the best transverse Mercator — 923 parts per million, exactly the optimum, with nothing left on the table. Scored over the box the Ordnance Survey’s own published extent suggests, 7.6° west to 1.8° east, the adopted grid is 1,842 against a best of 1,280, and the winner is not a transverse Mercator at all but an oblique one with its axis turned twenty degrees.

Three declarations of the same country, three different verdicts about the same grid.

None of the three is wrong, and the variation is not noise. A region is an input to this problem in exactly the way a projection is, and a box is a declaration somebody makes. The optimum central meridian is, to a good approximation, the middle of the declared region — so the question is the adopted grid optimal? is not answerable without a statement of what the country is taken to be.

This site does not have coastlines, and the reason is the founding decision: a coastline dataset has a simplification level, so a figure computed from one is partly a measurement of the vendor’s generalisation. The consequence here is that the region has to be declared, and the consequence of that is visible above. It would be equally true with a coastline: the answer would then depend on whether islands, territorial waters and offshore work were inside it.

What was computed, and how #

Sixty-odd candidates, each scored the same way.

Each candidate is evaluated at 26 × 26 sample points over the region; the largest principal scale and the smallest are recorded; the answer is their ratio. The transverse and oblique members are built by the site’s general aspect machinery rather than by a bespoke formula, so an oblique Mercator here is the same object every other essay would build. The conics run across their cone constant. The stereographic is centred on the region.

The bound comes from the least-squares solver, given the region’s boundary in a stereographic chart centred on it and asked for the conformal map whose scale is constant there. Its own residual on the boundary is 1.0 × 10⁻⁵, which is the fit’s accuracy rather than the answer’s.

The whole comparison is done on the sphere, deliberately and with the cost stated. The Chebyshev machinery solves on the sphere, and mixing a spherical bound with ellipsoidal candidates would compare two different problems. What the flattening does to a scale spread over a country is second order and is measured elsewhere on this site; what a projection choice does to it is what this figure is about.

What the bound leaves out #

The gap between 1,066 and 1,442 parts per million is real, and closing it would mean adopting a map with no closed form — a series of coefficients solved for one country, evaluated by summing a polynomial, with no name and no textbook.

That is a genuine option today and was not one in 1936. It is also, on this evidence, not worth taking. The difference between the optimum and the best named projection is 376 parts per million in a ratio, which over a hundred-kilometre line is 38 millimetres of scale spread — against a grid whose adopted scale factor already deliberately introduces four hundred parts per million of distortion in order to halve the worst case.

So what the bound buys is not a better grid. It is the ability to say how much of the remaining error is the projection’s fault and how much is the region’s, which is a statement no comparison between two candidates can make.

The one region whose answer has a name #

The solver used above is trusted for one reason, and it is not that it converges. For a cap the answer is known in closed form: the least-spread conformal map of a spherical cap is the stereographic projection centred on it, and the spread is sec⁴(ρ/2). That is the case the machinery is verified against, to seven parts in 10¹⁴, before it is pointed at a shape with no closed form.

A country is not a cap, which is precisely why a general solver is needed and why the answer for Britain has no name. The gap between the two situations is the whole reason this rung could not have been written earlier: a criterion whose answer is a projection with a name is a fact to quote, and a criterion whose answer is a table of coefficients is a computation to run.

Where the worst point is #

There is a reason the criterion is stated on the boundary and it is the same reason the spread can be optimised at all. The logarithm of a conformal map’s scale factor is a harmonic function, so it obeys the maximum principle: its extremes over a closed region are on the edge of it. Where the worst point is establishes that for the largest scale error on a conformal map of a country, whatever the country’s shape and whichever conformal projection was chosen.

That turns an optimisation over a region into a condition on a curve, which is what makes the fit a linear least-squares problem rather than a search. It is also why the sensitivity of the previous section is so sharp: the answer depends on the boundary and on nothing inside it, so moving the declared boundary is not a small perturbation of the problem — it is the problem.

What a line actually pays #

The reason the spread is the design quantity, rather than an abstraction, is what a surveyor does with it. A distance measured on the ground is reduced to the grid by a scale factor that is a property of a point, and a distance is not measured at a point — so the factor has to be averaged along the line, and the three rules in use for that differ by a hundred metres on a 362-kilometre baseline.

Everything in that difficulty is proportional to how much the scale factor varies over the region. A grid with half the spread would halve the disagreement between the averaging rules, halve the correction itself, and halve the length at which the correction becomes worth making at all. The parts per million in the bar chart above are not an aesthetic quantity: they are the size of the arithmetic every job in the country has to do.

Where the model stops #

Four things this measurement does not include, each of which a real design must.

A grid has to be computable. The transverse Mercator survived because four coefficients and a hand calculator produce it; a solved conformal map is a table of coefficients that has to be distributed, versioned and implemented identically in every piece of software that touches the country’s data.

A grid has to be stable. Once adopted, every coordinate in the country refers to it, and the argument against a marginally better map is the one a published coordinate is a result makes: changing the definition moves every published number.

A grid is often a zone system. Where two zones meet is about the seam, and a country large enough to need several zones has a different optimisation — how many, and where — rather than this one.

And the scale spread is not the only criterion. It is the right one for distance work, which is what a grid is for, but a projection that minimises it need not minimise the area error, and nothing here is claimed about what a hectare on the grid is worth on the ground.

The generalisation #

The structure is one every applied optimisation has and few state: there is a bound, there is the best member of the family anybody would actually use, and there is what was adopted — and the three gaps mean different things. The gap from the adopted map to the best named one is a design decision that could be revisited. The gap from the best named one to the bound is the cost of insisting on a formula with a name. The bound itself is a property of the region and of nothing else.

Reporting only the first is what a comparison between two candidates does, and it makes a difference of a few hundred parts per million look either negligible or damning depending on which two candidates were chosen.

Who found it, and when #

Chebyshev stated the criterion in 1856 and Grave proved it in 1896, and it has been the one optimality theorem in the subject ever since. What it has rarely been is used: the proof is for a region with a smooth boundary and the construction needs a conformal map to be solved for, which until recently meant a boundary-value problem nobody was going to run for a national grid.

Britain’s grid was defined in 1936 and its constants were chosen by exactly the reasoning the shallow curve above justifies: a central meridian near the middle of the country, and a scale factor chosen to halve the worst case. The finding that it is the optimum for one declaration of the region and not for another is a fact about the declaration, and it would have been unavailable to anybody who could not compute the bound.

The bound’s real use is as a certificate for the grid that exists #

A theorem that has rarely been used invites the reading that it should now be used to design something. That is the wrong conclusion here, and the shallow curve is the reason: a grid designed against the bound would differ from the grid a committee chose by a few hundred parts per million, which is not a case for replacing a national coordinate system.

What the bound does support is the opposite claim, and nobody could previously make it. Britain’s grid was defended in 1936 by an argument about reasonableness — a central meridian near the middle, a scale factor halving the worst case — and that argument establishes that the choice is sensible. It cannot establish that no other choice is much better, because much better is a statement about everything that was not tried.

The bound is a statement about everything that was not tried. Computing it says how far the existing grid sits from the best a conformal projection of that region could do, and the answer being small is a certificate rather than a coincidence.

And a certificate is worth more than a design here, because the cost structure is asymmetric. Redesigning a national grid costs the rewriting of every dataset, every map sheet, every survey record and every piece of software that names it, against a saving of a few hundred parts per million. Confirming that the existing one is near-optimal costs a boundary-value problem and settles the question permanently.

A certificate is also the thing that can be handed to somebody proposing a replacement.

It also says which questions remain open. The bound is computed against a declared region, and the finding above is that Britain’s grid is optimal for one declaration and not for another. So the certificate is conditional on a boundary somebody has to state — and that, rather than the optimisation, is where the remaining judgement lives.

Where the ladder goes next #

This rung optimises a map for a region whose shape is declared. The other half of the practical problem is what happens to a measurement once the grid exists — how a slope distance on the ground becomes a grid distance, and what happens to that chain when the observation is a satellite baseline rather than a tape. That is the chain the satellite does not have, and it is where the design question stops and the working question starts.