The shortest route between two coasts

Every rung of this anchor takes two points and finds a curve between them. The points are given — London and Tokyo, a departure and a destination — and the whole question is which curve.

Nothing anybody actually does starts that way. A ship sails from a country to a country, a cable lands somewhere on one coast and somewhere on another, a flight connects two regions with airports in them. The two ends are choices, and choosing them is a question this anchor has never asked.

It has an answer, and the answer is not visible on a map.

Why the answer is on the boundaries #

The search would be over four numbers if the two points could be anywhere in their regions, and it is over two, because the minimum is on the edges.

The argument is one sentence. If the nearest pair had a point strictly inside one region, the route could be shortened by moving that point towards the other region — and moving it a little way keeps it inside, because it started strictly inside. So it was not the nearest pair. Therefore the nearest pair is on the boundaries, for any two regions that do not overlap.

That is enough to make the problem finite and it does not make it easy. The objective is a distance between two points running along two closed curves, it has no derivative shortcut worth having, and it has local minima wherever the two boundaries have facing bulges. Every number here comes from a coarse sweep over all 480 × 480 pairs of sampled boundary points followed by a pattern search in both boundary parameters, which is what stops the refinement settling on the wrong bulge.

The three quantities a region pair has #

Once both ends are free, a route acquires quantities a two-point route does not have, and separating them is most of what the rung is for.

The length of the shortest route. One number, the minimum over both boundaries, and the thing everybody thinks they are asking about.

Where the pair is. Two points on two edges, and the quantity that actually decides where a cable lands or a ferry runs. It is not recoverable from the length.

How flat the minimum is. How much longer the second-best pair is, and how far from the first. A sharp minimum means the pair is decided; a shallow one means the length is decided and the pair is not, so any other consideration — a harbour, a border, a depth — settles it at almost no cost in distance.

Only the first is a number a route essay would normally report. The second is what a page gets wrong, and the third is what decides how badly.

The page picks a different pair #

A reader of a map does not run a search. They look, and they pick the two places that look nearest, and what looks nearest is a page distance.

The two columns say different things and the second is the one that matters.

Fourteen per cent on a length is a bad answer and it is a recognisable kind of bad answer — the sort of error every projection in this collection makes about every distance, and one a reader who knows the map is aware of. Two thousand one hundred and thirty-five kilometres on a position is a different failure. The map has not stretched the answer; it has named two different places.

For the Atlantic pair the true nearest points are the north-west corner of the European box, at 10° west and 64.9° north, and the eastern edge of the American one at 66° west. A cylindrical page puts the pair much further south, because a cylindrical page stretches the high latitudes and makes the northern route look long — which is exactly the distortion the shortest route is not straight is about, arriving as a choice of endpoints rather than as a curve.

Where it is worst, and it is not where the region is largest #

Three pairs of regions give three quite different answers, and the pattern is not about size.

Europe to the United States: true 3,580.62 kilometres, worst page 536.96 long, 15.0 per cent.

Japan to New Zealand: true 8,406.75 kilometres, worst page 66.84 long, 0.795 per cent — twenty times better as a fraction, on a route twice as long.

Chile to New Zealand: true 7,069.80 kilometres, worst page 1,588.90 long, 22.5 per cent, with the chosen pair 1,716 kilometres away from the right one.

What decides it is not the size of the regions but how much the pair moves as the page changes, and that depends on how flat the objective is near its minimum. Two compact regions facing each other across a gap have a sharp minimum and every page finds nearly the same pair. Two long thin regions running past each other — Chile is 39 degrees of latitude and 10 of longitude — have a shallow minimum along the whole facing edge, and a page that tilts the objective by a few per cent moves the answer a long way down it.

That is a general property of a shallow optimum rather than anything about maps, and it is the same shape the shape of the valley measures for a projection-choosing search: the depth of a minimum decides how much it matters and the width of it decides how far a perturbation moves the answer.

Why the gnomonic is not the right page for this #

The gnomonic is the projection this anchor keeps returning to, because it is the only one that draws every great circle straight. The gnomonic companion is the essay about it, and a reader who has followed the anchor would expect it to win here.

It does not, and the reason separates two properties that are usually run together.

Drawing a great circle straight is a statement about the shape of a route. It says the route between two given points is a straight line on the page, which is what makes a gnomonic chart useful for laying one out. It says nothing at all about the route’s length: the gnomonic’s radial scale grows as sec²ρ, so a route twice as far from the centre draws several times too long.

What this rung needs is the second property and not the first. The question is which of many pairs is shortest, so the page must rank lengths correctly, and a page that draws every route straight while getting all their lengths wrong ranks them wrongly. An azimuthal equidistant page centred on the pair does better because its radial scale is exactly one, so a length through the centre is a true length.

Neither is exact for a wide gap, because both are exact only about routes through the centre and these routes are chords rather than radii. What would be exact is a page on which every pair’s separation is its ground distance, and four cities that cannot be drawn to scale is the essay showing there is no such page.

A page distance is not a monotone function of a ground distance #

The failure has a compact statement and it is worth having, because it explains why every page fails and why some fail worse.

A page is a good ranking instrument for lengths if longer on the ground implies longer on the page. That is a global monotonicity requirement and no projection has it — a projection’s scale factor varies over the sheet, so a short route through a stretched region can draw longer than a long route through a compressed one.

Which is not the same as being a bad map. A projection with an areal factor of nine and an angular deformation of forty degrees can still rank two nearby routes correctly, because both are distorted by nearly the same amount. What breaks the ranking is the scale factor changing between the two candidates, and that is a property of the region they span rather than of the projection’s worst case.

So the right question about a page used for this is not “how distorted is it” but “how much does its scale change across the strip the candidate pairs lie in”. Distortion over a region is the essay about turning a pointwise distortion into a regional number, and the quantity this rung needs is not any of the criteria it lists — it is the spread of the areal or linear factor over a strip, and no published index reports it.

What the refusal is #

A measurement in which no page finds the right pair is a measurement whose search is broken, and one in which every page finds it is a measurement of nothing. Both halves are checked.

The check that the search is right is that three azimuthal pages centred on the true pair’s midpoint find that pair to a part in ten thousand on the Atlantic case. They should: an azimuthal projection has unit scale at its own centre in every direction, so a neighbourhood of the centre is drawn to scale, and a page drawn to scale ranks lengths correctly.

The check that the failure is real is that the cylindricals are out by more than twenty times the best centred page on every pair tried, which is the general form of the assertion — exactness is required only where the gap is narrow enough for it to hold.

The same question, asked of many regions at once #

Two regions have one shortest route. Several have a network of them, and the object that results is one this collection already owns from the other end.

Nearest of many is a partition divides a surface among several sites by which is nearest, and the boundaries of that division are the places where two sites tie. Between regions rather than points, the same construction gives a division of the surface by which region is nearest — and the ties are curves, the meeting points are tripoints, and every one of them is decided by a minimum over a pair of boundaries rather than by a distance between two points.

Nothing in this collection computes that, and the reason is worth stating rather than left as an omission: the partition machinery takes a list of sites and asks a distance of each, which is one evaluation per site per place, and a region needs a minimisation over its whole boundary instead. On the sampling used here that is a factor of four hundred and eighty, which is affordable for two regions and not for a grid over the sphere times seven of them.

What is affordable, and is what a real system does, is to approximate each region by its nearest boundary point to the query — which is the same minimisation, done once per query rather than once per pair. The error that introduces is not measured here.

What this changes about the anchor’s own figures #

Every route in this anchor is drawn between two named places, and none of the numbers moves. What moves is the status of the choice of places.

Why Mercator exists measures the rhumb line’s excess over the great circle for a set of routes between named cities. The flattening cost does the same for a set of pairs. In both, the pairs are given and the measurement is about the curve between them.

If the pairs had instead been chosen as “the nearest crossing between these two landmasses”, every one of them would carry the error above — and it would be an error in the inputs rather than in the calculation, so no amount of care with the geodesic would show it. That is the general hazard: a computation done exactly on inputs chosen by eye off a page inherits the page’s distortion, silently, and reports a number with too many digits.

The defence is cheap and is the one this rung uses. Choose the pair on the ground, by search, and then draw it on whatever page the reader needs.

Where the true pair actually sits, and why nobody would guess it #

The Atlantic pair is worth reading off, because its position is the whole content of the rung and it is not where anybody puts a finger.

The shortest crossing between the European box and the American one runs from 10° west, 64.9° north — the north-west corner of the European region, which is in the Norwegian Sea north of Iceland’s latitude — to 66° west on the American region’s eastern edge, which is off the Maritimes. It is 3,580.62 kilometres.

Nothing about either end is a landmark. Neither is a corner of a real coastline, neither is a port, and the European end is a corner of a bounding box rather than of anything geographic — which is a limitation of the region library and not of the method. What the position does show is that the crossing is a northern one, which is the same fact every transatlantic route in this anchor turns on and which every cylindrical page conceals.

A reader looking at a Mercator chart of the North Atlantic sees Greenland the size of Africa and the northern crossing correspondingly enormous, and picks a pair much further south — 2,135 kilometres further south. The cable and shipping routes that actually cross the North Atlantic go north, and they were laid by people who did the arithmetic rather than by people who looked.

What a real coastline would change #

The regions here are boxes and ellipses, which is what this collection’s region library holds, and a real coastline is neither.

Two things would change and they pull in opposite directions. A real boundary is far more complicated, so the objective has many more local minima and the coarse sweep would need to be finer — the search gets harder. And a real boundary is rougher, so the true nearest pair is a headland facing a headland and the minimum is sharper rather than shallower, which makes the answer more stable against a page’s distortion.

Which of the two wins is not something this rung can say, because this collection has no coastline data and does not want any. What it can say is that the failure it measures is a property of shallow minima, and that a shape with a long facing edge is where to look for one.

The one number a page does get right #

There is a quantity in this rung that every page reports correctly, and finding it is worth as much as finding the failure.

The length of the route between the pair the reader chose is exactly what a route calculation gives, on any page, because the calculation is done on the ground once the two ends are named. A reader who picks the wrong pair off a Mercator chart and then asks for the great-circle distance between those two places gets 4,085.30 kilometres, and that is the true length of the route they have chosen. Nothing is wrong with the arithmetic. What is wrong is the choice it was handed.

That separation matters because it says where the defence has to go. Improving the distance calculation does nothing — it is already exact. Improving the projection helps a little and cannot fix it, since no page ranks all lengths correctly. What fixes it is doing the choice on the ground, which costs one search and is the whole of the remedy.

It is the same lesson the line drawn straight on the page is a route reaches from the other side: a curve chosen on a page and realised on the ground is a real curve with a real length, and the length is not the problem — what the page decided is.

The route as a single decision #

Every rung of this anchor, including this one, treats a route as something chosen once. Two points are fixed, or a pair is searched for, and then a curve is found between them and that is the answer.

Nothing anybody actually flies or sails is like that. A crossing is re-planned as the weather changes, so what gets flown is a chain of shortest paths from a sequence of moving starting points, and whether that chain is anywhere near the shortest path from the original start is a question no single optimisation answers.