Designing a grid for one region

Every country in the world is covered by Universal Transverse Mercator. Most countries publish a national grid anyway. The question that decision turns on is how much better a grid designed for one place can be, and it has an answer.

Two parameters, nested #

A transverse Mercator grid for a given ellipsoid has two free parameters that affect distortion: where to put the central meridian, and what scale factor to apply. The false origin and the true origin’s latitude do not affect distortion at all — the first because it is an origin that is not there, the second because a transverse Mercator’s scale factor does not depend on the northing.

So the optimisation is two-dimensional, and it separates. For any candidate meridian, the best scale factor has the closed form $1/\sqrt{k_{\max}}$ derived in the scale factor was chosen. That reduces the problem to a one-dimensional sweep over the meridian, with the inner problem solved exactly at every step.

The nesting is worth naming because it is the reason this is computable at all. A joint search over both parameters would be a two-dimensional minimax with a non-smooth objective; a sweep over one parameter with a closed form for the other is a curve that can be drawn.

What the curve’s shape says #

The sweep over meridians is a V as well, and for a different reason from the scale-factor V.

The scale factor’s V comes from a maximum swapping between two candidates. The meridian’s V comes from the region’s own geometry: move the meridian west and the eastern edge gets further away, move it east and the western edge does. Both edges’ distances grow linearly with the displacement, and the worst case follows whichever is further — so the minimum sits where the two edges are equidistant from the meridian, which for a symmetric region is its middle.

Britain’s optimum at 2.9°W is not quite the midpoint of the sampled box, which is at 2.9°W by construction. That the two agree exactly here is a consequence of the box being a box; over a real outline the optimum shifts toward whichever side carries more ground at the latitudes where a degree of longitude is shortest, which is the north.

That last point is a real asymmetry and it is worth stating: the cost of being far from the meridian is measured in metres, not in degrees, so a country’s northern extremities pull the optimal meridian toward them more strongly than its southern ones. It is the same effect that makes UTM’s constant scale factor stop helping above 57.5°, seen from the design side rather than the operational one.

The answer for Britain #

The best central meridian is 2.9°W with a scale factor of 0.9992995, giving a worst-case distortion of 703 parts per million over the region sampled.

UTM zone 30, whose central meridian is at 3°W, manages 1,059 parts per million over the same ground.

The improvement is a factor of 1.51.

Why 1.51 and not something impressive #

The number is modest, and understanding why is more useful than the number.

Britain is about nine degrees of longitude wide, and a UTM zone is six. So the region does not fit inside a zone at all, and the comparison above is between a purpose-built grid and a zone applied outside its intended width. That should favour the purpose-built grid heavily, and it barely does.

The reason is that the best meridian for Britain — 2.9°W — is almost exactly UTM zone 30’s meridian at 3°W. The international zoning scheme, which was chosen with no reference to any country, happens to put a boundary in a good place for this one. All the purpose-built grid gets to improve is the scale factor, and improving the scale factor buys a factor of two at most, which is where 1.51 comes from once the meridian’s small advantage is added.

For a country the zoning scheme treats less kindly the factor is much larger. A country straddling a zone boundary must either use two zones — with all that where two zones meet records about the seam — or use one zone far outside its width, at a cost that grows quadratically.

What a country actually buys #

The distortion factor is not the main reason countries publish grids, and the essay would be misleading if it stopped here.

A national grid buys one zone instead of two or three, which removes the seam problem entirely for domestic work. It buys a coordinate system whose numbers are recognisable — a British six-figure grid reference is a cultural object, not merely a technical one. And historically it bought a grid computed on the national datum, which mattered enormously when transforming between datums meant a table lookup rather than a computation.

The distortion improvement is real, measurable, and third on that list. Presenting it as the reason is a common move in the literature and it does not survive being measured.

The narrow-zone alternative #

There is a completely different design that gives up on covering a country with one grid.

Several American states publish low-distortion projections — dozens of small zones, each tuned to a county or a corridor, each with a scale factor chosen so that the combined factor is one at the local mean elevation. Wisconsin, Oregon, Minnesota and Colorado all have such systems.

Those grids are worse by the classical measure and better by the only one that matters to their users: a distance measured on the ground equals a distance computed from the coordinates, with no reduction at all. The projection has been made deliberately wrong so that the reduction becomes unnecessary — the argument the ground is not the grid sets out, and it is the same trade this collection keeps meeting in different clothes.

The cost is arithmetic and administrative rather than geometric: fifty zones for a state means fifty sets of parameters, fifty opportunities to use the wrong one, and a seam wherever two meet. The narrower the purpose, the better the fit — and the more of the fit’s benefit is spent on managing the boundaries.

Running it for a country the zones treat badly #

Britain’s 1.51 is a lower bound on what a national grid can be worth, because the zoning happens to suit it.

A country that straddles a zone boundary faces a different arithmetic. Using one zone means carrying its extent far outside the six degrees the scale factor was chosen for, and the cost grows as the square of the excess: a region twelve degrees wide handled in one zone has four times the worst-case distortion of one six degrees wide, before any optimisation. Using two zones means a seam through the middle of the country, with everything where two zones meet records about what crossing it costs.

Against either of those, a purpose-built single grid with a well-placed meridian and an optimised scale factor is worth a great deal more than 1.51 — which is why the countries with the most emphatic national grids tend to be the ones the international scheme cuts through rather than the ones it happens to fit.

A grid is also a promise #

The last thing worth saying about designing a grid is that the design is the smallest part of publishing one.

A national grid is a commitment to maintain a set of parameters, a datum, a transformation to the global system, and a network of marked points, for as long as anybody uses the coordinates — which in practice means forever, because a published coordinate is a result that legal descriptions come to depend on. Britain’s grid has outlived the datum it was computed on, in the sense that OSGB36 is now defined by a transformation from a satellite-based system rather than by the triangulation that produced it.

So the optimisation in this essay is a genuine input to the decision and is nowhere near the whole of it. A country choosing between its own grid and the international zones is choosing between a factor of one and a half in distortion and a century of institutional obligation, and the second consideration is the larger one.

What the optimisation cannot see #

Every number above is a distortion figure, and a distortion figure is a statement about the map. A survey’s actual difficulty is a statement about the reduction, and the two do not rank grids the same way.

A grid with a scale factor of 0.9993 is excellent by the classical measure and requires every ground distance to be shortened by seven hundred parts per million before it becomes a coordinate — seven metres in ten kilometres, on every line, forever. A grid with a scale factor of 1.0004 is worse by the classical measure and, at 2,551 metres of elevation, requires no reduction at all, because the grid factor and the elevation factor cancel. That cancellation is the ground is not the grid, and no optimisation in this essay knows about it.

So there are two design objectives here that are usually conflated, and they pull in opposite directions over most of a zone. Minimising distortion pushes the scale factor below one; making the reduction vanish pushes it above one by whatever the local elevation requires. A designer must choose, and no published national grid the collection has looked at states which choice it made.

What was computed, and how #

A sweep of 41 candidate central meridians, spanning two degrees either side of the region’s midpoint.

At each, the inner optimisation: the closed form for the scale factor, confirmed by a 401-point search, over a 21×21 sample of the region.

The UTM comparison, computed the same way over the same region rather than taken from the zone’s nominal specification — because a zone’s published worst case is stated for a zone six degrees wide at the equator and the region here is neither.

The assertion. The designed grid must beat the zone. That is a weak-looking check and it is the right one: a stronger assertion naming a factor would have to be updated whenever the sampled region changed, and would then be asserting the region rather than the geometry.

The sensitivity, which decides how much precision the answer needs #

An optimum is only worth stating to the precision at which it matters, and the curve says how much that is.

The meridian curve is a V with steep sides, so a meridian a degree away from the optimum costs a real amount — a few hundred parts per million over Britain. The scale-factor curve is also a V, but the outer sweep has already optimised the scale factor at every meridian, so a tenth of a degree of meridian error is nearly free.

The consequence is that the ten-digit scale factors national grids publish are not precision for its own sake but are also not doing what they appear to. The scale factor must be stated exactly because it multiplies every coordinate and any truncation is a systematic error; its optimality is flat enough that the seventh digit is irrelevant to distortion. Those are two different reasons for two different kinds of exactness, and conflating them is how “0.9996012717” acquires a reputation for spurious precision it does not deserve.

There is a ceiling on what any amount of parameter-fitting can buy, and it follows from the same picture. The optimisation moves the zone’s scale curve up or down bodily; it cannot change the curve’s shape, which is fixed by the projection. That is why the improvements here are factors rather than orders of magnitude, and why a country that wants an order of magnitude has to change the projection rather than tune the one it has.

Where the model stops #

The region is a box and a country is not. Optimising over Britain’s true outline would move both parameters, and the collection’s standing refusal to use a coastline dataset applies — a shapefile has a generalisation level, so an optimum computed from one is partly a measurement of the vendor’s simplification. The box is stated as an approximation.

The objective is the worst case. Minimising the mean, or the population-weighted mean, gives different answers, and the catalogue of what each choice encodes is distortion over a region. No optimisation here is more principled than its objective, and choosing the objective is the step that turns a measurement into a judgement.

The family is fixed. Everything here searches within transverse Mercator. A Lambert conformal conic would be better for a region wider than it is tall, and an oblique Mercator better for one lying along a diagonal — the point of the aspect is a free choice. Searching across families as well as within one is a harder problem and this essay does not solve it.

And the true optimum over an arbitrary region is not known. Chebyshev’s criterion says the best conformal projection of a region is the one whose scale factor is constant on the boundary, and for a spherical cap that projection is known exactly. For a country-shaped region it is a boundary-value problem this collection does not solve, which Chebyshev’s criterion states rather than skirts. What is computed here is the best transverse Mercator, which is a much smaller claim.

It is worth saying where the 1.51 sits against its own ceiling, because the ceiling is a round number.

A tangent transverse Mercator has a worst-case departure of kₘₐₓ − 1 over its region. Applying the optimal scale factor 1/√kₘₐₓ leaves √kₘₐₓ − 1, which is half of it to first order. So choosing a scale factor optimally, rather than not choosing one at all, buys a factor of exactly two — no more, at any region size, for any transverse Mercator.

Britain’s national grid gets 1.51 of that two, and the missing quarter is not waste: UTM already applies a scale factor of 0.9996, which is some choice and is not the best one for a region nine degrees wide. Splitting the ceiling accordingly, UTM’s fixed constant is worth about 1.32 of the available two for Britain and the national grid’s tuned constant is worth the remaining 1.51.

That decomposition is the honest form of the essay’s own point. There is no unclaimed factor lying about: the total available from parameter choice is two, most projections in service already take part of it, and a purpose-built grid is competing for the remainder. Anything larger than a factor of two has to come from somewhere else — a better-placed meridian, a different projection family, or a country the zoning scheme cuts through.

The generalisation #

Two nested optimisations, where the inner one has a closed form, are a different kind of problem from a joint search — and recognising that a problem separates is usually worth more than any amount of cleverness about how to search it.

The second lesson is about how modest improvements get reported. A factor of 1.51 is worth having and it is not the transformative gain that “a grid designed for this country” suggests. The honest version of the claim needs the number attached, and the number is only available to somebody willing to run the comparison — which is the collection’s whole method applied to a design decision rather than to a projection.

Who did it, and when #

Martin Hotine chose Britain’s parameters in the 1930s, with no computer, by an argument he could carry out by hand. That he landed within a tenth of a degree of the meridian a modern search finds is the most striking thing in this essay: the closed form was available to him, the region was a judgement then as now, and the arithmetic that took a sweep of 41 meridians here took him an afternoon and a slide rule.

UTM’s zoning was fixed by the US Army in the 1940s and inherits from a nineteenth-century German scheme. Its designers were explicitly not optimising for any country — the specification’s whole purpose was that a soldier anywhere in the world could be given the same instructions — and the fact that zone 30 happens to suit Britain is a coincidence that flatters the international system rather than the national one.

Where this goes next #

Many small zones is the other design, and its cost is at the boundaries: where two zones meet. Where the tuning goes all the way to the ground rather than to the ellipsoid, the result stops being a map projection at all — a grid scaled to the ground is not a map.