A reach set with a cost that depends on direction

A circle of a distance is not a circle draws a range ring and finds it 4.17 times longer in one direction than another on a common projection. Nearest of many is a partition divides the sphere by distance. A corridor has a width the page cannot keep buffers a route by one.

All three build their sets out of a distance, and a distance has two properties nothing anybody travels has: it is the same in both directions, and it is the same in every direction.

What is being computed, and why it is not a radius #

The set of places within a stated distance is a ball: walk out along every great circle and stop at the same angular radius. That is the construction the three rungs below this one use, and it needs no algorithm.

The set of places within a stated time is the solution of a Zermelo problem. The cost of a step is its ground length divided by the speed made good — the vehicle’s own speed plus the component of the flow along the step — so a step downwind is cheaper than the same step upwind, and the cheapest route to a place is not the great circle to it.

That is a shortest-path problem rather than a formula, and it is computed here by a wavefront: a grid over the sphere, sixteen neighbours per node, and Dijkstra outward from the source. The output is a time to every place rather than a rim, and the rim is a level set of it — which is why the set can have any shape at all rather than the one a construction was built to give it.

The lattice, and the thing it nearly hid #

A shortest-path algorithm on a grid does not measure the metric it was given. It measures the best route made of the steps available, and with eight steps per node the cheapest route to a place twenty-two and a half degrees off an axis is a staircase, which is longer than the straight line.

On this grid, with eight neighbours, the still-air reach set comes out with an axis ratio of 1.10 — a ten per cent anisotropy with no flow in it at all.

That is most of the size of the effect the first figures were drawn to show, and it would have been invisible: an eight-lobed rosette with a ten per cent amplitude looks exactly like a plausible physical result. Adding the eight knight’s moves takes it to 1.057, and interpolating the rim rather than reading it off the nearest node takes it to 1.023.

Two point three per cent is this measurement’s floor and every number below is quoted against it. The still-air set also comes out at 2,966 kilometres where the truth is 3,000 — one per cent short, in the direction a lattice always errs.

How large the effect is #

The jet’s shape is the interesting one. Its flow is strong on a single parallel and nearly absent elsewhere, so the cheapest long route is to climb to that parallel, run along it and come back — which makes the reach set a pair of lobes rather than an ellipse, and makes the rim a function with two maxima. A reach set under a structured flow does not merely stretch; it changes its topology of extrema, and no scaling of a circle produces that.

The gyre is the quiet row and is quiet for a reason worth naming — and it is quiet at 1.15, which is still five times the lattice’s floor, so quiet does not mean absent: a circulating flow helps as much as it hinders on any closed excursion, so its effect on a reach is second order. Its effect on the route is not, and that is the quickest route is not the shortest, one anchor along.

What the flow costs, and what it does not #

That is the finding this rung is for. A flow changes which ground is reachable enormously and how much of it hardly at all.

At a flow of 0.45 the set is 2.58 times longer one way than the other and covers 1.10 times the ground. At 0.75 it is 5.16 times longer and covers 1.28 times the ground.

An isotropic model would predict the opposite of that. If a flow simply made the vehicle faster on average, the set would grow and stay round; if it made it slower, the set would shrink and stay round. It does neither: the gain downwind and the loss upwind nearly cancel in area, and do not cancel at all in shape.

The practical form is blunt. An operator who plans with a range ring and a mean speed will have the right amount of ground and the wrong ground, and the size of the error is the axis ratio rather than the area ratio.

That also explains why the mistake survives. A planner who checks their range ring against the ground actually covered will find it right to ten per cent, which is well inside anybody’s tolerance, and will never learn that the set is wrong by a factor of two and a half. Area is the summary that is easy to validate and it is the one the failure does not touch — the same shape of problem an average of ellipses is not an ellipse has, where a mean that is right hides an object that is wrong.

The other flows, drawn #

The plane’s answer, which the area column is measured against #

There is an exact result for the simplest of the four flows and it sharpens the finding rather than replacing it.

In a plane, under a flow that is genuinely uniform, the reach set in a stated time is a circle of the vehicle’s own radius vT, translated by the drift fvT. That is immediate: the vehicle’s own motion can take it anywhere within vT of wherever the water has carried it, and the water has carried it a fixed vector. So

$$\text{ratio} = \frac{1+f}{1-f}, \qquad \text{area} = \pi v^2T^2, \ \text{exactly}.$$

At f = 0.45 the ratio is 2.636 against a measured 2.58, which is the agreement the lattice’s one-per-cent shortfall predicts. And the area does not change at all — not by a little, exactly not at all, because translating a circle does not change its area.

That is a stronger version of the essay’s own finding. A flow changes which ground is reachable enormously and how much of it hardly at all is an understatement in the planar case: the area change is zero, and the whole of the effect is a displacement.

So the 1.10 and 1.28 in the area column are not the flow’s doing in the sense the planar theorem describes. They are the sphere’s: a steady westerly is a constant speed along parallels, and parallels are not parallel lines — a vehicle carried east is carried along a curve whose convergence changes what its own circle can reach. The area growth is a curvature effect riding on a shape effect that would exist on a flat sea.

Two things follow. The planar formula is the right first estimate for a uniform flow and it is available without any computation: an operator who knows the flow as a fraction of their own speed knows the axis ratio exactly. And the area, which is the statistic easiest to validate and the one a planner is most likely to check, is guaranteed by a theorem to be uninformative — it is constant in the case where the shape error is largest and best understood.

The jet and the gyre have no such formula, which is why they are in the catalogue. But the row that anchors the whole rung is the one where the answer is known in advance and the machinery has to find it.

Separating the two anisotropies #

There are now two things that can make a drawn reach set elliptical, and this collection has only ever had one of them.

The projection’s own, which rung 1 measures: a range ring 4.17 times longer one way than another on the Lambert cylindrical, entirely because of the map — and 13.03 times at 70° north, which is larger than anything a flow does here.

The cost’s, which is on the ground and would be there on a globe.

Every figure in this rung is drawn in an azimuthal equidistant centred on the source, where every radius from the centre is a true ground distance — so the shapes above carry none of the first. That is not a convenience; it is the only way to state the second at all, and it is the same control the strain a map adds to the ground’s needs for a different quantity.

On any other projection the two multiply, and a reader shown a reach set on an ordinary sheet is being shown a product of two anisotropies with no way to factor it.

What the check refuses #

Every number here rests on a still-air control and the control is the whole of the method.

The still set has to be round. The same lattice, the same wavefront, the same rim-finding, with the flow switched off, must return a rim that is the same in every direction. It returns 1.023, and the departure from one is the lattice’s, not the cost’s. That number is quoted beside every ratio because a ratio of 1.15 — the gyre’s — is only seven times the floor.

And the still set has to be the right size. It comes back at 2,966 kilometres against a stated 3,000, one per cent short, which is the amount a lattice always loses to the staircase. A control that was round and the wrong size would mean the cost model was wrong; one that is the right size and not round would mean the lattice was. Both have to hold, and both do.

The two together are why the eight-neighbour version was thrown away rather than reported with a caveat. Its still set was round to ten per cent, and ten per cent is the middle of the range of effects this rung is about.

Which flows are worth stating #

Three stated fields carry the whole of this rung, and it is worth saying why those three rather than a catalogue.

A uniform flow is the case with an exact answer to compare against: reversing it is a reflection, so the forward and backward sets are congruent, and any measurement that said otherwise would be a measurement of the lattice. Everything the next rung does rests on that symmetry.

A jet is the case where the reach set changes shape rather than merely stretching. A vehicle’s cheapest long route climbs to the fast parallel and runs along it, so the set acquires a second maximum, and no summary as an ellipse can carry that. It is the field that shows a reach set is a solution rather than a formula.

And a gyre is the quiet control. A circulating flow helps as much as it hinders over any closed excursion, so its effect on a reach is second order — 1.15 against the lattice’s own 1.02 — while its effect on a route is first order and large. A field that changes routes and not reaches is exactly the case that separates the two questions, and having one in the catalogue is what stops the anchor from being a single measurement repeated.

One consequence for the picture: a reach set under a flow has no radius, so the range ring that every operational chart draws is not an approximation to it but a different object. The ring answers how far could this go if nothing helped or hindered, which is a question about the vehicle. The set answers where could it get to, which is a question about the vehicle and the medium together, and the two are the same only in still air.

Where the model stops #

The cost is first-order Zermelo: the speed made good is the vehicle’s plus the flow’s component along the step. That is exact when the flow is slower than the vehicle and is what every navigation rule of thumb uses; the full problem steers into the flow rather than along the step, which buys a little more and changes the shape by less than the lattice’s own floor at these strengths.

The flow is steady in time. A reach set under a flow that changes while the vehicle is travelling is a different object — the wavefront has to be integrated forward with the field — and nothing here approaches it.

And the grid is a lattice, with the consequences priced above. The floor is 2.3 per cent and the figures quote it; an effect of that size measured with this machinery would not be a measurement.

What a chart could show instead #

The reach set is a shape and the thing charts print is a radius, so three repairs are available and they cost different amounts.

Print the rim. The set is a closed curve and drawing it costs no more ink than a circle does; what it costs is a computation per chart rather than a compass. That is the honest product and it is what a modern flight-planning system produces, under the name of a fuel-range polygon.

Print two radii and a bearing. An ellipse fitted to the rim is a poor description of the jet’s two-lobed set and a good one of the steady flow’s, and it is a great deal better than one number. It is also the same object as Tissot’s ellipse — two principal values and an orientation — which is a coincidence of shape rather than of subject and is a convenient one.

Or print the worst case. The shortest reach, in the least favourable direction, is one number and is the one an operator planning a return trip actually needs. It is 1,653 kilometres where the mean-speed circle says 3,000, and a chart that printed it would be conservative rather than wrong.

None of the three is what a range ring is. A range ring is the mean, drawn as though it were the set, and a mean is not a member of the family it summarises.

The generalisation #

A set defined by a cost is only a ball when the cost is a metric, and most costs are not metrics.

A metric needs symmetry and a triangle inequality. Travel time under a flow keeps the triangle inequality and loses the symmetry, which makes it a quasi-metric — and a quasi-metric’s balls are not round, are not the same forwards and backwards, and cannot be summarised by a radius.

That covers a great deal more than navigation: a travel-time isochrone on a road network with one-way streets, a cost surface with a slope, a diffusion in a drift. In every case the temptation is to fit a radius and report it, and in every case the quantity a radius throws away is the direction.

Who found it, and when #

Zermelo posed the navigation problem in 1931 — the fastest route for a vessel in a current of known strength — and its solution is one of the founding examples in the calculus of variations. The reach set is the level set of its value function, and the machinery to compute one on a grid is Dijkstra’s, from 1959, or its continuum cousin the fast marching method.

What is unusual here is not the computation but the comparison. Reach sets in cartography are drawn as circles because the classical treatment of them is a treatment of distance, and the two literatures — optimal control and map projection — have not much needed each other. The measurement this rung reports is what happens when the second lends its instrument to the first.

Where the ladder goes next #

The moment the cost stops being symmetric, within a stated time of here and within a stated time of there stop being the same question. The next rung measures how far apart the two answers are, and the answer is most of them.