Choosing for a line, not a region

Every regional criterion in this subject is an integral over an area, weighted by the area element. That is the right instrument for a country and the wrong one for a pipeline.

Two different questions #

An area objective asks: over this region, how far from unity do the scale factors depart, averaged with the area element? It is answered by an integral over two dimensions, and its answer is a projection whose distortion is spread evenly over the region.

A line objective asks: along this curve, how far from constant is the scale factor? It is answered by an integral over one dimension, and its answer is a projection whose distortion is banished sideways, off the curve, into ground where there is nothing to measure.

Those are different problems with different answers, and the second one is what a great many surveys actually face. A pipeline is 1,000 km long and 50 m wide. A railway, a canal, a power line, a coastal chart and a river survey are all the same shape. So is a flight corridor and so is a fibre route.

The area objective answers the wrong question for all of them, and it does so quietly: the projection it returns is perfectly reasonable, its distortion is well distributed, and along the one line that matters it is fifteen per cent out.

The answer, and why it is obvious once stated #

Mercator has one line of true scale: its own equator. The aspect is a free choice, so that line can be put anywhere — including along the corridor.

The construction is one rotation. Take the great circle through the two ends of the route, find its pole by a cross product, and build a Mercator projection whose axis points at that pole. The rotated equator is the corridor, so the corridor is exactly at true scale, and the scale grows away from it in both directions like $\sec$ of the distance — which is to say, over a 50 m wide survey strip, not at all.

Measured over the Madrid–Tokyo corridor:

Two features of that table are worth separating. The oblique Mercator’s 0.03 parts per million is not a good result; it is a zero, achieved by construction, with the residual being the numerical noise of the derivative machinery. And the best ordinary alternative is out by a factor of five million more.

Why this is not a tautology #

An oblique Mercator laid along a corridor is true along the corridor by construction, and pointing that out and declaring victory would be a circular argument.

What makes it a claim is the second half: the same projection is not the answer to the area question over the same ground. Score the region around the corridor by an area criterion and the oblique Mercator is unremarkable — it has been optimised along a line and has bought that with distortion everywhere else, and the area criterion counts everywhere else.

So the two objectives genuinely disagree about which projection to use, which is the thing the essay is asserting. The gate requires both halves: the oblique Mercator must hold the corridor to under 100 ppm, and the best ordinary alternative must be worse than 10,000 ppm, or the corridor objective would not be a distinct question.

That second threshold is where the measurement had something to teach. The first version compared the two by dividing one scale spread by the other, and a scale spread is a number near one: a projection 15% out over the route came out as “1.2× worse” than one that is exact. Comparing in parts per million instead of as a ratio of ratios turns 1.2 into five million, and the earlier framing would have hidden a real failure behind a comfortable-looking number.

What the map looks like, and why nobody minds #

An oblique Mercator fitted to a corridor is a strange-looking map. The graticule runs diagonally, the poles are somewhere off to the side, and continents at right angles to the route are unrecognisable.

None of that matters, and saying why is the whole argument for purpose before property. A corridor survey’s map is a working document for people measuring a strip of ground. It never shows a continent. Its sheet is a few kilometres wide and hundreds long, and the only quantities read off it are distances and bearings within that strip. A projection judged by how the world looks on it is being judged by a criterion nobody in that room is applying.

Drawn on the normal Mercator instead, the same pair of routes gives a familiar-looking world, a great circle that is a curve rather than a line, and a scale that varies by 119 per cent along it. A corridor survey would trade all of the third for either of the first two.

The generalisation, and where it stops #

The construction generalises to any route on a great circle, and every corridor of a few hundred kilometres is close enough to one.

Where it stops is a route that is not: a river, a coastline, a border following a watershed. Those curves have no great circle to be laid along, so no single oblique Mercator is true along the whole of them, and the design problem becomes the general one — a projection minimising the scale variation along an arbitrary curve.

That problem has the same shape as the minimum-distortion problem for an arbitrary region that this site has deferred twice, and for the same reason: it is a boundary-value problem rather than a formula. What is available in the meantime is the practical answer surveys actually use — break the corridor into segments, fit an oblique Mercator to each, and carry the discontinuity at the joins, which is the zone system’s answer applied to a line rather than to a globe.

The rivals change places, which is the second finding #

That last figure is the corridor version of the observation the regional essay makes about countries: a ranking of projections is a statement about a specific case rather than a property of the list.

Over Madrid to Tokyo — an east–west route at mid-latitude — the Lambert conformal conic is the best of the ordinary projections at 15%, because a conic’s line of true scale runs east–west and the route roughly follows it. Over Cape Town to London — a north–south route crossing the equator — the same conic is the worst at 131%, because its standard parallels are now perpendicular to the corridor.

So the useful advice is not use a conic for corridors. It is: find the projection whose line of true scale runs along the route, which for an arbitrary route means constructing one rather than choosing one.

What it looks like as a distortion pattern #

The oblique Mercator’s virtue and its cost are the same fact seen from two directions, and a distortion field shows both at once.

Drawn as a whole sheet, the band of low distortion follows the route across Eurasia and everywhere else on the map pays for it — and at the two rotated poles, which are ordinary points of the Earth, the projection runs away entirely.

The two rotated poles are the interesting part. On a normal Mercator the singularities are at the geographic poles, where a reader expects trouble and where a map is usually cut off. Rotate the projection and the singularities move to two ordinary points on the equator of the corridor’s great circle — in this case in the South Atlantic and the North Pacific — where nothing about the geography warns of them.

That is a hazard specific to oblique aspects and it is worth stating: rotating a projection rotates its singularities too, onto ground that has no reason to look dangerous. A corridor survey never goes near them, and a system that reuses the corridor projection for anything else might.

What this makes of the classical grids #

Two national grids were built on exactly this reasoning and are usually described as curiosities.

Laborde’s projection for Madagascar, 1928, is an oblique Mercator with its line of true scale running along the island’s long axis, which is diagonal. Madagascar is a corridor — 1,600 km long and 570 km wide — and a conventional transverse Mercator would have needed either an implausibly wide zone or a boundary down the middle of the country.

The Swiss grid, from Rosenmund’s 1903 design, is an oblique Mercator of the same kind fitted to Switzerland’s east–west extent.

Both are usually presented as national eccentricities. They are the corridor objective, solved correctly, by people who had the problem and did the arithmetic — and the reason there are only a handful of them is that most countries are closer to round than to long, so most countries face the area problem rather than this one.

The area answer over the same ground, for comparison #

Running the ordinary area criterion over the region containing the corridor is what most software would do, and the answer it gives is worth putting beside the corridor answer.

The comparison is not that one ranking is right and the other wrong. It is that they are answers to different questions, and the mistake worth avoiding is asking the first and using the answer for the second. Every projection minimises something; a projection chosen for a corridor by an area criterion is minimising something the job does not care about.

Why the objective is almost never stated #

A survey specification says the coordinate system shall be UTM zone 34N or shall be the national grid, and stops. What it does not say is what objective that choice serves, and in almost every case the objective is compatibility rather than accuracy: everybody else’s data is in that system, so this data must be too.

That is a legitimate objective and it is a different one from either of the two in this essay. Naming the purpose before the property means noticing that interoperability is on the list of purposes, that it usually wins, and that it is what makes the corridor question academic for most projects — the corridor projection would be better and the data would be unusable.

Where the corridor objective does win is exactly where the data is not shared: the construction survey itself, where the coordinates are internal to one job, exist for a year, and are converted to the national grid at the end. That conversion is a known transformation with a known scale factor, so nothing is lost, and the survey gets to work in a system in which its own line is true.

The width the construction actually buys #

One number turns the argument from a geometric curiosity into an engineering decision: how wide the strip can be before the oblique Mercator’s own distortion matters.

Away from its line of true scale, Mercator’s scale factor is $\sec$ of the angular distance, so at a distance $w$ from the corridor the scale is larger by about $w^2/2R^2$. Setting that against a tolerance gives the same quadratic law the plane-survey limit turns on, with a different constant:

So a corridor projection at 10 parts per million is exact enough over a strip 57 km wide, which is more than any pipeline survey needs and is why the construction is not a delicate one. A corridor of any realistic width sits entirely inside the region where the projection is true to a part in $10^5$.

That also settles what to do when the route wanders. A river meandering 20 km either side of its overall great circle is still inside the 10 ppm strip, so the oblique Mercator fitted to the endpoints serves the whole survey — and the general boundary-value problem the previous section deferred turns out not to arise for any corridor narrow enough to be called one.

A third corridor, at a different angle again, to show that the construction does not depend on the route running any particular way.

How much ground the construction actually buys #

The half-width table can be set against the best a compact map can do, and the comparison says exactly what the corridor objective is worth rather than merely that it is worth something.

Both laws are square roots and both come from the same quadratic. The corridor’s usable half-width is w = R√(2τ); the least scale spread any conformal map of a cap can achieve gives a usable radius of r = 2R√τ for the same tolerance. Dividing,

$$\frac{r}{w} = \frac{2R\sqrt\tau}{R\sqrt{2\tau}} = \sqrt2.$$

Sideways, the corridor projection is worse than an optimally centred cap by a factor of 1.414, at every tolerance, and the two laws are otherwise identical. So the oblique construction buys nothing at all in the direction across the route — it buys everything in the direction along it, where a cap has a boundary and a corridor does not.

Turning that into ground is what makes the trade concrete. At ten parts per million a cap reaches 28.5 kilometres and covers 2,550 square kilometres. A corridor strip at the same tolerance is 57 kilometres wide and runs for as long as the route does: over a thousand kilometres it covers 57,000 square kilometres, twenty-two times as much ground held to the same scale tolerance, and the ratio grows in proportion to the route’s length.

That is the whole case for the corridor objective in one number, and it explains why the construction is worth a rotation rather than being an elegance. A survey that can accept 28.5 kilometres in every direction should use a well-centred conformal map and stop; a survey that needs a thousand kilometres in one direction cannot, at any tolerance, and the oblique Mercator is not a better answer to its question but the only one.

What was computed here #

The corridor is the great circle between two places, sampled at 81 points. Each candidate projection’s principal scale factors were evaluated at every point by the site’s derivative machinery and normalised by the geometric mean over the route, so that the comparison is of variation rather than of overall size.

The oblique candidate is built with the library’s own aspect, rotating the sphere so that the pole of the corridor’s great circle becomes the projection’s pole. That pole is a cross product of the two endpoints’ unit vectors, and the construction throws rather than returning nonsense if the two points are coincident or antipodal — which is the degenerate case the paths ladder spends a rung on.

Three claims are asserted: the oblique construction wins along the corridor; it holds the scale to better than 100 ppm; and the best ordinary alternative is worse than 10,000 ppm.

What the pictures cannot show #

Each curve is normalised to its own mean, so the figure shows variation and not size. Two projections with identical variation and a factor of two between their overall scales are indistinguishable here, correctly — a constant scale factor is a known number a survey divides out.

The corridor maps also cannot show the width of a corridor. A survey strip is a few kilometres across and the sheets here are the whole world, so the region within which the oblique Mercator is essentially exact is thinner than the line drawn for the route.

Who found it, and when #

Hotine formalised the oblique Mercator in 1946–47 in a series of papers on the geometry of the geodesic, and the construction is now usually called the Hotine oblique Mercator after him. Rosenmund and Laborde had each built one for a specific country decades earlier without a general treatment.

The objective itself is older than any of them and belongs to engineering rather than to cartography: a canal survey of the eighteenth century needed the scale along the cut to be right and did not care what happened either side, and solved it by working in a local system fitted to the line. What the twentieth century added was the observation that such a system is a projection, with a formula, that can be published.

Where this goes next #

Both fourth rungs on the choosing ladder have been about matching the map to the job. What remains on the practical side is what a grid does to the two quantities a survey actually measures, which are a distance and an angle.