A corridor has a width the page cannot keep

The first two rungs of this ladder took reach from a point. A great deal of reach is taken from a line: everywhere within so many kilometres of a pipeline, a flight corridor, a coastline, a road. In a spatial database the operation is called a buffer, it is one function call, and what it returns is a polygon that looks entirely unremarkable.

It is a reach set like the others — the partition rung divided the whole surface and this one takes a strip out of it — and it fails on a page in two ways that have nothing to do with each other. One is about how it is drawn. The other is about how its area is computed, and it is the more surprising of the two, because it goes wrong in a formula rather than in a picture.

The set, and its closed form #

The set within a distance w of a geodesic segment of length L on a sphere of radius R is a tube with two rounded ends, and its area is

$$2LR\,\sin(w/R) \;+\; 2\pi R^2\,(1 - \cos(w/R))$$

— the band, plus two half-discs that make one whole one. The planar answer that every buffer implicitly assumes is 2Lw + πw², and the two agree in the limit because R sin(w/R) → w.

The check here is the one this site always makes: the closed form is compared against the set itself, sampled cell by cell with the exact closed-form cell area. For the London–Tokyo route at 200 kilometres the two agree to 0.02 per cent, which is the sampler’s own resolution and not a disagreement.

The gap between the sphere’s answer and the plane’s is small at ordinary widths, and it is worth writing down precisely because it is the part that sounds like it ought to matter and does not:

A 200-kilometre buffer along a route most of the way round the world is 0.016 per cent smaller than the plane says. That is not the problem with buffers, and saying so first is the point: the curvature correction to the formula is negligible at every width anyone buffers at. What is not negligible is everything else.

The same corridor, drawn where the failure is visible #

Drawing the same set on a second projection separates the two things a reader might be seeing. The corridor in the hero and the corridor here are the same ground: same route, same 200 kilometres, same 3.9485 million square kilometres. What differs is entirely the page.

A reader who takes the Mercator picture at face value concludes that the corridor is wider in Siberia, and there is nothing on the page to say otherwise — no scale bar helps, because a scale bar is right in one place and the corridor spans a hundred and forty degrees of longitude. This is the honest version of the picture. The dishonest version is the one in the previous section, where the set was never computed at all and the drawn width is a page constant pretending to be a ground constant.

The stroke and the set are two different objects #

Almost no corridor a reader sees was computed as a set. It was stroked: the route was drawn as a line, and the line was given a width. A drawing program has one width per stroke, so whatever ground distance that width means at one point of the route is what it means at every point.

Measured back on the ground along the London–Tokyo route, with the page width chosen to mean 200 kilometres at the midpoint:

Nothing is constant. The best is the plate carrée at 1.22 and the worst the Lambert cylindrical at 3.17, where one drawn stroke means 66 kilometres at one end of the route and 210 at the other.

This is the same mechanism as a road drawn two pixels wide, which found that a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north, and would need to be 23 pixels wide at 85° to mean the same thing. The difference is what is being claimed. A road’s stroke width is a drawing convention nobody reads as a measurement. A corridor’s width is the measurement — it is the whole content of the picture — and the page has no way to say that it is not keeping it.

Conformality does not rescue this either #

Mercator is conformal and comes eighth of ten. That is not an anomaly and it is the same reason the previous rung found: conformality is a statement about two directions at one point, and this is a statement about two points. The stroke at London and the stroke at the Kamchatka end are the same page length; the scales there differ by a factor of 2.47; and no amount of local angle preservation constrains the ratio of scales at places 9,000 kilometres apart.

What would fix it is a projection whose scale is constant along the route, and that projection exists for a route rather than for a region: an oblique aspect with the route as its own equator, on a projection that is true along that line. That is a real answer, it is a different map for every corridor, and it is what the choosing ladder means by naming the purpose before the property. It is also why reporting the map rather than the parameters is the right convention for a corridor: the map that keeps the width is defined by the route, so quoting its parameters tells a reader nothing they can check.

The other way the width is stated wrongly #

There is a third way to ask for a corridor and it is the commonest of all: within 0.5 degrees of the route. It is available in every tool, it needs no projection, and it is not a distance at all.

A degree is not a unit of length measured what that request produces: on the ground it is an ellipse, 1,106 metres north–south against 558 east–west at 60° for a hundredth of a degree, and not a circle even on the equator, because the meridian’s radius of curvature is smaller than the parallel’s. A corridor asked for in degrees therefore has a ground width that varies with latitude and with the route’s own bearing, which is two variations rather than the one the stroked corridor has.

The relevant point for this ladder is that all three requests — a set within w kilometres, a stroke of p page units, a buffer of d degrees — produce different regions, and are written down in the same sentence. Only the first is a reach set. The other two are page and coordinate objects that resemble one, and the resemblance is close enough at small extents that nothing complains.

What the corridor is mostly made of #

The two terms of the closed form behave differently and the difference has a consequence that is easy to state and easy to miss.

The band term is proportional to L and grows nearly linearly in the width. The cap term does not contain L at all — the two ends of any corridor are a disc of its own width, whether the route is a kilometre long or most of the way round the world. So the ends are second order in the width and the middle is first order, and past some width the ends win.

For London–Tokyo the caps are 1.62 per cent of the set at a 100-kilometre width and overtake the band at 6,000 kilometres. For a route a tenth as long the crossover would be a tenth as far out. Anything called a corridor at a width comparable to its own length is not a corridor, it is two discs joined by a strip, and no name in the toolkit distinguishes them.

The crossover has a closed form, which is worth extracting because the figure reads it off a ladder of widths and the expression says what it depends on. Setting the band equal to the caps gives L sin x = πR(1 − cos x) with x = w/R, and the half-angle identities collapse that to

$$\tan\frac{w}{2R} = \frac{L}{\pi R}, \qquad w = 2R\arctan\frac{L}{\pi R}.$$

For the London–Tokyo route, at 9,559 kilometres on a sphere of 6,371, that is 5,685 kilometres — which is where the ladder puts it, between its 4,000 and 6,000 steps.

In the planar limit the same balance is 2Lw = πw², so w = 2L/π, and for this route that would be 6,085 kilometres. The two differ by seven per cent, and the direction is the one the rest of this essay leads to expect: on a sphere the band term saturates as sin(w/R) turns over while the cap term keeps growing, so the ends win sooner than the plane says.

The closed form also corrects the scaling remark above. Halving the route halves the crossover is exactly true in the planar limit, where the crossover is 2L/π and nothing else, and it is only approximately true on a sphere, where the arctangent flattens. A route a tenth as long has a crossover almost exactly a tenth as far out, because the argument of the arctangent is then small; a route twice as long does not have a crossover twice as far, because the arctangent is bounded by π/2 and the crossover is therefore bounded by πR — half the circumference, whatever the route. No corridor on a sphere is more band than end past πR, however long the route it follows, which is a statement the planar formula cannot make and which the saturation table above is the numerical face of.

Where the formula stops being the area #

Push the width out and the closed form fails — not by drifting, but by counting ground twice.

The two agree to a part in ten thousand out to 10,000 kilometres, which is most of a quarter of the circumference. At 12,000 they part. At 15,000 the formula returns more area than the sphere has, and at 18,000 the sampled set has saturated at 510.07 million square kilometres — the whole world, exactly — while the formula is still rising.

The mechanism is the same one the previous ladder met at the cut locus. Walk outward from the route in both directions and the two fronts eventually meet on the far side, and past that meeting every place is being counted by both. The formula has no way to know: it is an integral of a width along a length, and a length has no idea what is happening at the antipode.

The fourth panel is the picture worth sitting with. What a very wide corridor leaves out is not near either endpoint — it is the two places farthest from the route as a whole, which sit off to the sides, roughly ninety degrees away. A corridor is a set, and a set’s complement has a shape that has nothing to do with the shape of the thing that generated it.

Why this is not a curiosity at working widths #

The saturation happens at widths nobody buffers at, which is exactly the reason it is worth stating. The formula does not announce that it has stopped being true. It returns a number, the number is plausible, and the only check that would catch it is to compare it against the area of the sphere — which is a check nothing performs, because nothing expects to need it.

The working-width version of the same failure is a self-intersecting buffer round a curved or branching route: a river with a meander tighter than the buffer width, a road network buffered as a set of separate lines. There the two fronts meet over a few kilometres rather than at the antipode, the closed form over-counts the overlap by exactly the doubly-covered area, and the number is out by a few per cent rather than by five per cent of the world. Same mechanism, same absence of a warning, and it is caught only by computing the union as a set rather than summing the formula per segment.

What a corridor is for, and what that implies #

A corridor is almost always a membership test: is this settlement within the flood zone, is this cable within the exclusion area, is this parcel within the road reserve. Which means the quantity that matters is not the area at all — it is whether the boundary is in the right place, place by place.

That reframes the three failures above by size. The curvature correction to the area moves the boundary by nothing at all: the closed form and the set have the same boundary, and only the arithmetic that totals them differs. The double counting also moves no boundary — it produces a wrong total for a correctly-bounded set. The stroked width is the only one of the three that moves the boundary, and it moves it by up to 110 kilometres on a route like this one, which is the difference between the widest and the meant width on Mercator.

So the ranking of the three failures by area error is nearly the reverse of the ranking by consequence, and this is a general hazard rather than a fact about corridors: an error reported as a percentage of a total is being reported in the units of the operation that is least likely to be the one anybody is performing.

The three failures, separated #

It is worth keeping the three apart, because they have different sizes and different fixes and they are habitually reported as one:

The curvature correction to the area is 0.016 per cent at a 200-kilometre width on a 9,559-kilometre route, and 0.38 per cent at 1,000 kilometres. It is real, it is computable in closed form, and it is almost never what matters.

The stroke width on the page is a factor of 1.22 to 3.17 along one route, and it is not a correction at all — it is a different object being drawn. The fix is to compute the set and project it, which is what every figure in this essay does, rather than to stroke the projected route.

The double counting is nothing until the fronts meet and then it is unbounded, up to the whole sphere. The fix is to compute the union rather than the sum, and the test is whether the answer exceeds the surface it is on.

Only the first is a curvature effect. The other two are what happens when a set is inferred from a curve — and the whole of this ladder is the difference between the two.

What the ladder has established #

Three rungs, three kinds of reach set, and one finding common to all of them: a page is much better at curves than at regions, and the operations that produce regions do not say so.

A range ring is drawn round only on the azimuthal family, and spaced correctly by exactly one member of it. A nearest-site partition hands away between 0.75 and 22.16 per cent of the ground depending on a plane nobody chose deliberately. A corridor is drawn at a width that varies by a factor of three along its own length, and its area formula eventually claims more world than there is.

None of those is a distortion measurement in the sense the rest of this collection uses. There is no indicatrix here, no angular deformation, no areal factor; the quantities are areas of misassignment, ratios of ground widths, and the point at which a formula stops describing a set. Tissot’s machinery answers what a map does to an infinitesimal circle at a point, and every question on this ladder is about something that has an extent — which is why the answers could not be read off the derivatives, and had to be measured on the sets themselves.