The rule that keeps a route near land is pinned at both ends

Every route in the essays before this one optimises a single thing. The shortest route is not straight minimises length; the quickest route is not the shortest minimises time in a moving medium; a crossing is a chain of decisions re-plans the quickest route as a forecast fails. Each objective is taken as given, and everything else about the route is free.

A twin-engined aircraft crossing an ocean is not free. It is permitted to fly only where some suitable airfield is within a stated distance — the distance it can cover on one engine in the time the rule allows — and that is a second objective of an entirely different kind. It is not a cost that adds up along the route. It is a region the route must stay inside: the union of a circle of the stated radius round every airfield.

Where a route inside circles can bend #

The route in that figure is not found by a search over a grid, and the reason it does not need one is worth giving first, because it is the whole of the method.

Inside a circle on a sphere — a spherical cap, smaller than a hemisphere — the great-circle arc between any two points stays inside the cap. So inside a union of caps a shortest route can only bend at a place where it is forced to: a corner of the union, where the rims of two caps cross and no third cap covers the crossing. At any other point on the route, the route could be straightened a little along a chord of whichever cap it is in, the chord would still be inside, and the route would be shorter.

That turns the problem into a finite one. List the corners — at 745 kilometres forty-three pairs of the twenty circles overlap, so there are at most eighty-six — join every pair whose connecting arc stays inside the union, and find the shortest chain from New York to London through them. Whether an arc stays inside also has a closed form: along a great circle the part inside any one cap is a single interval of distance, so the arc is inside the union exactly when those intervals cover it end to end.

A route that must go round solves the opposite problem — a route forbidden to enter one circle — and finds two tangent arcs and a stretch of rim. The difference is instructive. A route kept out of a circle hugs its rim, because the rim is the shortest way round a convex obstacle. A route kept inside a union of circles never follows a rim at all; it runs in straight arcs from corner to corner, because from the inside every rim bulges away.

One gap decides whether the ocean can be crossed #

Shrink the radius and the corners pinch together until, at some radius, the union of circles stops connecting New York to London and no route is permitted at all. That least radius can be found by shrinking and routing, and it can also be found without routing anything.

Two circles of equal radius overlap exactly when their centres are closer than twice the radius, so the union connects two airfields exactly when some chain of airfields joins them with every link shorter than twice the radius. The least radius is therefore half the widest link of the best chain — the chain whose worst gap is as small as possible — and that is a classical problem with a classical answer. It is the path through the tree that joins every airfield by the shortest total set of links, and its longest link is the bottleneck.

The two methods agree: bisection on whether a route exists gives 624.66 kilometres, and the widest link of the chain gives 624.66, to a metre. The whole North Atlantic crossing is decided by one gap, between Labrador and the southern tip of Greenland. Every other link on the chain could lengthen by half again and the least radius would not move by a metre, and no amount of airfield-building anywhere else would lower it.

That is a minimax quantity rather than a sum, and it behaves like one. It is insensitive to everything except the worst link, which is why it can be read off a map of airfield positions by anyone with a ruler and the patience to find the chain — and why a change to one airstrip on one coast can matter for a whole ocean while a dozen new airfields in the wrong places matter for nothing.

One point decides where the rule stops costing anything #

The upper wall is the mirror of the lower one, and it is set by a single point as well — a point on the great circle rather than a gap between airfields.

The great circle is permitted exactly when every point of it is within the radius of some airfield, so the radius at which the rule stops costing anything is the furthest the great circle ever gets from its nearest airfield. That is 1,250.2 kilometres, reached once, in mid-ocean between the southern tip of Greenland and Iceland and far to the south of both.

The corners in the profile have a meaning of their own. Each is a place where the nearest airfield changes — from Gander to Narsarsuaq, say — which is a crossing of the boundary of the partition of the ocean among its airfields. The peak sits on one of those boundaries, as it must. Along a stretch of track with a single nearest airfield, the distance to it cannot rise to a maximum in the middle and fall again, so the furthest the track gets from land is where the nearest airfield hands over to the next.

Finding that point exactly took more than sampling it. Sampled every ten kilometres, the peak reads 1,249.8 kilometres and names Keflavík alone, because every sample lies a little nearer one of the two airfields than the other; the true peak lies between two samples, where the distances are equal. A lower envelope of distances has its maximum at a crossing, and a sampled maximum of one is always a little low and always names only one side of the crossing.

There is a coincidence in the two walls that deserves reporting and not explaining. Twice the least radius is 1,249.3 kilometres and the free radius is 1,250.2 — within a kilometre of each other. They are set by different features of the ocean, a gap between Labrador and Greenland and a point equidistant from Greenland and Iceland, and although Narsarsuaq happens to take part in both, nothing in the construction relates the two numbers. With another pair of cities or another list of airfields they would not agree.

The price between the walls has no steps #

Between 624.66 kilometres and 1,249.8 the rule costs something, and the natural expectation is a staircase. A route is stuck in one corridor of overlapping circles until the radius grows enough for two circles that were apart to touch; then a shorter corridor opens, and the length drops.

The staircase does not appear, and it is worth being exact about why, because it is not guaranteed. A step would need a corridor that is shorter than the one in use from the very moment it opens. When two circles first touch, the only way through the new gap is through the single touching point, and a route forced through one point is usually longer than the corridor it might replace. As the radius grows the gap widens, the route through it straightens, and at some radius its length falls through the length of the corridor in use — at which point the shortest route changes corridor with the two lengths exactly equal.

Between the two walls fifty-four pairs of circles begin to overlap, and not one of them opens a corridor shorter than the one in use. Every change of corridor is a crossing of two lengths, and the length is continuous through all five: ten metres of radius either side of each differ by what the slope of the curve predicts and by nothing more. A different list of airfields could contain a gap whose new corridor is shorter from the moment it opens, and that list would have a step; this one does not, and nothing here says how rare such a gap is.

What the changes do produce is corners. The shortest length is the lesser of two curves that cross, and where two curves cross the lesser has a kink.

This curve is also the answer to a question a crossing is a chain of decisions left open: what the set of routes looks like when there are two objectives that disagree. Every point on the curve is a route that cannot be made shorter without going further from land, or kept nearer to land without being made longer. The projections that are beaten on both counts finds the same kind of object among world maps and calls it a front; here the front is a curve with a wall at each end, which is a shape the projection front, drawn over a finite library, could not have.

The first kilometre of allowance buys the most #

The curve is steepest at its lower wall, and it is not merely steep there.

A square root has an infinite slope at zero, so the rate at which allowance buys route is unbounded at the wall. The first kilometre of extra radius beyond 624.66 is worth 11.9 kilometres of route; the tenth kilometre is worth 1.9; a kilometre of allowance at 1,000 is worth 270 metres, and at 1,200 it is worth fifty.

The mechanism is geometric and short. At the least radius the circles round Goose Bay and Narsarsuaq touch at one point, and the route must pass through it. Enlarge the radius by ε and the two circles overlap in a lens whose width across the gap is $\sqrt{2r\varepsilon}$ to leading order — the ordinary geometry of two circles that have just begun to intersect. The route can now cross the gap anywhere in that width, and moving the crossing point sideways changes the route’s length in proportion to how far it moves. So the saving is proportional to the lens’s width, which is proportional to $\sqrt{\varepsilon}$.

That argument also says when the law would fail. If the route happened to cross the gap at right angles — heading straight through the touching point — moving the crossing sideways would change its length only at second order, and the saving would grow linearly instead. The route here meets the gap obliquely, heading north-east across the Labrador Sea, which is why the exponent is a half. It is a property of this crossing’s geometry rather than of every such rule.

The practical reading is blunt. A rule set just above the least radius is set where every kilometre of it is most expensive. An aircraft permitted 630 kilometres from an airfield flies a route 262 kilometres long of the great circle; one permitted 660 flies a route 43 kilometres shorter than that, for thirty kilometres of extra allowance. The same thirty kilometres added at 1,000 would save under eight.

Four allowances, four ways across #

Read left to right and top to bottom, the four panels show the route migrating south. At the least radius it is pinned to the chain of airfields along the Greenland coast and through Iceland and the Faroes, which is the only chain whose gaps are small enough. As the allowance grows it lets go of airfields one at a time — Goose Bay, then Kulusuk, then Vágar — and swings towards the great circle, until at 1,000 kilometres a single crossing of circles south of Greenland is all that holds it north of the direct line.

Each of those release points is one of the corners on the price curve, and each is a place where two routes of equal length exchange places. A reader following the migration on the panels sees a route jump from one corridor to another; the length it has when it jumps is the same on both sides.

The rule, checked a second way #

The route was found by a method built on corners and intervals, and a check that used the same method would share its mistakes. So every route is also sampled every three kilometres along its length and each sample measured against all twenty airfields by plain distance.

Two facts have to come out and both do. No route strays past its allowance anywhere along its length, to the metre. And every route that bends reaches its allowance exactly — its furthest point from land is on the rim of some circle — which is what a shortest route inside circles must do, since a route that never touched a rim could be straightened.

A third check is built into the ends. With an allowance of five thousand kilometres, which puts the whole ocean in reach of land, the route found is the great circle to a millionth of a kilometre; and one kilometre below the least radius the method refuses to return any route at all rather than a least-bad one.

What the rule is, as a map-maker sees it #

The circles in these figures are range rings, and the page they are drawn on matters in the way a circle of a distance is not a circle describes. The conformal conic used here draws a circle on the ground as very nearly a circle, and its scale stays within a few per cent across the whole window — 1.1 per cent small at Goose Bay, 2.8 per cent large at Santa Maria — so every ring is drawn close to its true size relative to the others.

A Mercator chart would not, and it would misrank exactly the quantity that decides everything. The gap between Goose Bay and Narsarsuaq is 1,249 kilometres on the ground; the gap between Santa Maria in the Azores and Lisbon is 1,419. On Mercator, which enlarges lengths by the secant of the latitude, the northern gap is drawn as though it were about 2,300 kilometres and the southern one about 1,800. The page reverses their order: the gap that is narrower on the ground and decisive is drawn half a thousand kilometres wider than the one that is broader and irrelevant.

A planner judging the bottleneck by eye on such a chart would look for it in the wrong ocean. It is a ground distance between two points, and it has to be measured on the ground.

The bottleneck has an understudy #

The least radius is decided by one gap, and it is tempting to read that as fragility: take away Narsarsuaq, which sits on the decisive link, and the crossing should become much harder.

It becomes harder by 0.7 kilometres. Without Narsarsuaq the best chain runs from Goose Bay north to Iqaluit on Baffin Island instead, a gap of 1,251 kilometres against 1,249, and the least radius rises from 624.66 to 625.38. The second-best chain was waiting almost exactly behind the first.

That is the ordinary behaviour of a minimax quantity rather than a special property of this ocean. The least radius is the widest link of the best chain, and when that chain is removed the answer is the widest link of the next best one — which, on a coast with airfields spaced the way these are, is rarely much wider. A bottleneck is decisive and it is not unique, and a rule-maker counting on one airfield to hold an ocean open should first find the understudy.

Where the model stops #

The Earth is a sphere and the airfields are points. An ellipsoid moves every distance here by a fraction of a per cent and would move the least radius by a few kilometres; it would not change which gap is the bottleneck or the shape of the curve.

The air is still. A rule written in minutes of one-engine flying time is a distance only when there is no wind, which is how such rules are in fact written. A real crossing is planned in whatever wind there is, and the rule and the route then live in different metrics.

Every airfield is always usable. Weather, runway length, fire cover and opening hours make a real list of suitable airfields shorter than the list of airfields, and different on different days. Removing Narsarsuaq alone would raise the least radius, because it sits on the bottleneck link.

And real crossings are flown on organised tracks that move each day with the jet stream and are shared by hundreds of aircraft. The route here is the shortest permitted route for one aircraft on an empty ocean, which is the object the rule constrains and not the one a controller assigns.

Still open: what the rule does when the air moves #

A diversion allowance is written in time, and a time is a distance only in still air. In a wind, the set of places from which an airfield can be reached within the allowed minutes is not a circle: it is the set that can reach the airfield under a moving medium, which the set that can be reached is not the set that can reach shows is a different shape from the set the airfield can reach, and far from round.

So in a wind every circle in these figures becomes a lopsided region stretched downwind of its airfield, the corners move, and the bottleneck may move with them — a gap that is the widest in still air could be closed by a tailwind towards Greenland and a different gap opened by a headwind off Iceland. Whether the two walls stay pinned to single features when the circles are replaced by those regions, and whether the square root at the lower wall survives, is a question the still-air rule cannot ask.