The reach set takes the shape of the roads

A circle of a distance is not a circle draws a range ring and finds a page stretching it by a factor of thirteen. A reach set with a cost that depends on direction bends the ring with a flow. A partition under a directed cost has two versions divides the whole sphere with it. In every one of them the ground itself is featureless: a kilometre costs the same wherever it lies, and whatever shapes the reach set is a property of the vehicle and of the air or water it moves through.

Nobody travels on featureless ground. A vehicle that can leave the road still goes faster on it, roads run in particular directions, and those directions turn out to be what a reach set on real ground is made of. The simplest structured ground there is — a square grid of roads over open country — is enough to show it, and it has a closed form.

What a long journey can average #

The shape in that figure is not an accident of five-kilometre spacing, and the argument for what it converges to is short enough to give whole.

Over any short stretch the vehicle is doing one of two things. It is on a road, moving along it at the road’s speed, or it is off the road, moving in whatever direction it likes at the ground’s speed. Over a long journey it can do both in any proportion — a stretch along one road, a cut across a field, a stretch along another road — and the displacement the whole journey achieves is the time-weighted average of the velocities it used.

The set of velocities that can be averaged in this way is the convex hull of the ones available: every mixture of two achievable velocities is achievable, and nothing outside the mixtures is. On a two-way grid the road velocities are four points — the road speed V east, west, north and south — whose hull is a diamond, and the ground velocities are a disc of radius v. So as the roads become dense against the distance travelled, the reach set in a time T converges to T times the convex hull of that diamond and that disc.

Two things are absent from that statement and both absences are the finding. The surface does not appear: nothing about the terrain enters except the two speeds. And the positions of the roads do not appear either, beyond the directions they run in. The reach set on structured ground is decided by the structure’s geometry, not by the ground.

The hull has two regimes, and they meet at the square root of two #

A diamond whose corners sit at distance V from its centre has an inscribed circle of radius $V/\sqrt{2}$ — the distance from the centre to the middle of an edge. That single number decides the whole shape.

If the ground speed is below $V/\sqrt{2}$, the ground disc sits entirely inside the road diamond. It adds nothing to the hull, and the limiting reach set is the diamond alone. If the ground speed is above $V/\sqrt{2}$, the disc pokes out through the middle of every edge, and the hull becomes four arcs of the disc joined to the diamond’s four corners by eight straight tangents.

Writing α for the angle arccos(v/V), which is where each tangent meets the disc, the area of the limiting set is

$$A = 2\left(\tfrac{\pi}{2} - 2\alpha\right) v^2 T^2 + 4 V v T^2 \sin\alpha,$$

which is $\pi V^2 T^2$ when the two speeds are equal and the roads do nothing, falls as the ground slows, and reaches exactly $2V^2T^2$ — the diamond’s area — at the moment v falls to $V/\sqrt{2}$. It stays at $2V^2T^2$ for every slower ground after that.

Past the square root of two the ground stops mattering #

The cleanest way to see the two regimes is to compare how far the set reaches along a road with how far it reaches along the diagonal between two roads.

Below the kink the ratio is simply the speed ratio, and the reason is visible in the hull. Along a road the set reaches VT, because the fastest thing to do along a road is drive it. Along the diagonal the disc is outside the diamond, so the fastest thing to do is leave the road and go straight, at speed v. The two reaches are VT and vT and their ratio is V/v.

Above the kink the diagonal is reached faster by driving: along one road for half the distance and along the other for the other half, a staircase of length $\sqrt{2}$ times the straight line at a speed more than $\sqrt{2}$ times the ground’s. The ratio is pinned at $\sqrt{2}$ and the ground speed has dropped out of the answer.

That second regime has a practical consequence that sounds wrong and is not. On roads four kilometres apart, doubling the ground speed from 20 to 40 km/h — with the roads at 80 — moves the reach along the diagonal from 69.1 kilometres to 70.0, a change of 1.3 per cent, and all of that change is the grid’s spacing rather than the ground. Once the roads are 41 per cent faster than the ground, making the ground faster buys nothing along any bearing the roads already serve by a staircase, which is every bearing except the few within a narrow fan of the roads themselves.

The ratio sitting slightly above $\sqrt{2}$ past the kink — 1.428 for roads twice as fast, 1.447 for four times — is the limit being approached rather than violated. At a finite spacing the diagonal cannot be followed by a perfect staircase to its very end: the last partial block has to be made up somehow, and whatever is left over buys less diagonal progress than the idealised staircase would. The shortfall is a fraction of one spacing and it shrinks as the grid tightens, which is the next measurement.

A disc, a rounded diamond, a diamond #

The four panels are the whole argument as pictures, and the progression in them is worth reading slowly because it does not go the way intuition expects.

The first panel is the refusal. Roads no faster than the ground are not roads, and the set they produce is the disc the ground produces on its own. The area is counted in whole lattice cells and is good to about a per cent — 31,113 against the disc’s 31,416 — which is the resolution of the count rather than anything about the roads.

The second panel is the regime nobody draws. At a speed ratio of 1.2 the set is neither a disc nor a diamond: four flattened arcs join four sharp corners, and the flat stretches run along the tangents from the corners to the disc. Its measured area of 23,961 square kilometres agrees with the closed form’s 23,973 to half a part in a thousand.

The third and fourth panels are nearly indistinguishable, and that is the point. Between a speed ratio of $\sqrt{2}$ and one of 3 the ground has become more than twice as slow, and the reach set has lost 4 per cent of its area — almost all of it to the finite spacing, which the approach to the diamond still pays.

Why the limit is approached from inside #

Every panel’s measured set lies inside its dashed hull, and that containment is not a feature of these four cases. It is guaranteed.

The hull is the set of displacements a journey can average. A real journey on a real grid is one particular way of averaging them, with constraints the limit does not have: it must actually reach a road before it can use one, and it must follow a road to a junction before it can turn. Constraints only remove options, so no journey on the grid reaches anywhere the hull does not. A measured reach set that poked outside its hull anywhere would be a lattice inventing travel — which is exactly how the hull serves as a check on the instrument, and how one defect described below was found.

How fast the grid’s texture fades #

The limit is a statement about roads that are dense against the distance travelled. Real roads have a spacing, and the question that decides whether the limit is any use is how quickly the reach set approaches it as that spacing falls.

The approach is first order: halve the spacing and the departure halves. That is a different law from the square law every reach set ever drawn is too small finds for a fan of bearings, and the difference is in what is being approximated. A fan approximates a smooth curve by chords, and a chord’s shortfall is quadratic in its length. A grid approximates a mixture of velocities by a particular route through a lattice of roads, and the shortfall is the detour that route makes — the walk to the nearest road and the overshoot to the nearest junction — which is proportional to the spacing itself.

The practical reading of the slope is a rule of thumb with a stated error. A reach that is twenty times the road spacing sits within about four per cent of the limit on average. A reach only two and a half times the spacing is a third short of it, and at that point the limit is a poor description and the actual roads have to be drawn.

That rule matters because it says which of two quite different kinds of map a planner is making. At a reach of many spacings — a regional response plan, a day’s drive — the hull is the right picture and the individual roads are texture. At a reach of one or two spacings — a walking distance in a town, the last few kilometres of a delivery — the individual roads are the picture and the hull says almost nothing.

What a ring at road speed claims #

The most common way to draw how far a service reaches is a circle: a radius equal to a speed times a time, drawn round a station or a depot. On a grid of fast roads that circle is wrong before any page is involved.

Two over π is the ratio of a square’s area to its circumscribed circle’s, and it arrives here without any fitting. A ring at road speed over-claims by 36.3 per cent of its own area on any grid of roads more than $\sqrt{2}$ times faster than the ground, and equivalently it claims 1.57 times the ground that can be reached. The over-claim is concentrated in four lobes on the diagonals, where the ring extends to VT and the set to $VT/\sqrt{2}$.

That error is of a different kind from the one a circle of a distance is not a circle is about. There the ring was right on the ground and a projection distorted it on the page. Here the ring is wrong on the ground and no projection can repair it: an azimuthal equidistant page centred on the crossing draws the ring as an exact circle at the correct radius, and the reach set as a diamond inside it, and both are faithful pictures of two different sets.

It also changes a partition. Nearest of many is a partition divides ground among several sites by distance and finds that computing the division on the wrong page misassigns up to a fifth of it. On a grid of roads the boundary between two stations’ catchments is not the perpendicular bisector the distance division draws; it is the set of places equally quick to reach, and under a diamond-shaped metric that boundary has straight segments at 45 degrees wherever two stations are offset diagonally. A catchment map drawn with circles on a road grid is drawing the wrong metric, which is a larger error than drawing the right metric on a poor page.

The shape belongs to the network #

The diamond is a property of roads that run in two directions. Add a second pair of road families at 45 degrees and the argument gives a different hull without any further work.

With eight road directions the velocity points bound a regular octagon, whose inscribed circle has radius V·cos 22.5°, which is 0.924V. So the threshold beyond which the ground stops mattering falls from $\sqrt{2}$ to 1/cos 22.5° = 1.082: on a network with roads in four directions, roads only eight per cent faster than the ground already make the ground’s speed irrelevant everywhere but in narrow fans between the road directions.

The general statement is the one the octagon makes visible. The limiting reach set is the convex hull of the road directions at road speed, together with the disc at ground speed. A network with roads in many directions approaches the disc from inside, and the most anisotropic network there can be is the one with the fewest directions — which is the square grid, the layout of planned towns and of the section roads that follow the land survey lines across much of the rural American Midwest.

That last example carries a complication the plane hides. A survey grid laid out along meridians and parallels is not a square grid on the Earth, because meridians converge; the American survey handles it by restarting the grid on correction lines, and the roads jog where it does. Over a region small enough for how small is flat enough to call it flat, which the reaches here all are, the grid is square to far better than the lattice resolves. Over a state it is not, and the diamond’s axes rotate slowly with longitude.

The anisotropy is on the ground, and it has no north #

A reader who has spent time with a reach set with a cost that depends on direction will recognise a non-round reach set and expect the same things of it. Most of them do not carry over.

That anisotropy came from a flow, so it had a direction — a place downwind was quick to reach and slow to return from — and the set that can be reached is not the set that can reach measures how different the outward and inward sets become. A road grid’s anisotropy has no direction in that sense. A road is as fast in one direction as the other, the diamond is symmetric under reversal, and the set a place can reach is exactly the set that can reach it.

What it has instead is axes, and the axes belong to the roads rather than to the compass. A grid laid out at 30 degrees to north has a diamond at 30 degrees to north, and no statement about the reach set can be made in terms of latitude and longitude at all.

The lattice’s own ripple, measured #

Every number above comes from a shortest-path computation on a square lattice in kilometres, and a lattice has an anisotropy of its own that has to be measured before anybody believes a diamond.

The lattice allows a step to any neighbour whose offsets are at most three cells in each direction and share no common factor, which is thirty-two directions with a widest gap of 18.4 degrees between neighbours. A route heading between two allowed directions has to zigzag, and the zigzag’s excess is 1/cos(9.2°) − 1, which is 1.3 per cent. The measurement returns 1.29, never long, always short, which is the right sign for a lattice that can only ever add to a route’s length.

The equal-speed roads are the refusal. A computation that reported any difference at all between open ground and a grid of roads that are no faster than it would be reporting roads that do not exist, and it reports none, at any bearing, to the last digit.

The road that was a kilometre wide #

A second hazard was found by the hull rather than by the control, and it is worth recording because it is easy to make and hard to see.

The natural way to put a road on a lattice is to mark a row of cells as road and let any step through road cells go at road speed. That makes the road as wide as a cell, and at a crossing the corner of a road row touches the corner of a road column — so a diagonal step from one to the other lies wholly on road cells and cuts the corner of every junction at road speed. On roads four kilometres apart over a one-kilometre lattice, that produced a diagonal reach of 78.6 kilometres against a hull of 70.7: eleven per cent beyond the limit no journey can exceed.

Nothing about the picture looked wrong. The set was a slightly fat diamond, which is a plausible shape. What exposed it was the containment argument above: a measured set outside its hull is impossible, so the lattice was inventing travel. A road here is a line, and only a step that runs along a road line is charged at road speed; the diagonal then lands at 69.1 kilometres, inside the hull by the finite-spacing shortfall, where it belongs.

Where the model stops #

The grid is perfect. Every road runs the whole way, every crossing is a junction, and there are no delays at junctions, no congestion and no one-way streets. Junction delays would shift the diagonal’s staircase against the straight line in the ground’s favour; a real network would move the threshold above $\sqrt{2}$ by a stated amount this model does not compute.

The ground can be entered anywhere. A vehicle leaves the road wherever it likes and rejoins wherever it likes. Real roads are fenced, hedged and embanked, and access happens at gates — which turns the ground between roads into a set of pockets reachable only at points. That is a barrier with openings rather than a grid of lines, and it is a different structure with a different failure.

The start is a crossing. A reach set from a point halfway along a block pays an access cost the crossing does not, and its limit is the same hull shifted by at most half a spacing — which is why the first-order law above has the spacing in it at all.

And the region is flat. The reaches here are at most a hundred and ten kilometres, over which a plane agrees with the sphere to far better than the lattice cell. On a continental scale the roads are not a square grid on the Earth and the hull argument has to be made on the sphere, locally, with its axes rotating as the grid does.

Still open: whether a network with a hierarchy has a limit at all #

Every grid here has one spacing and one road speed. A real network has several of each at once: motorways fifty kilometres apart at one speed, main roads ten kilometres apart at another, lanes a kilometre apart at a third, and the ground between all of them.

The hull argument survives that — the limiting set is still the convex hull of every road direction at its own speed together with the ground disc, since a long journey can mix all of them. What does not obviously survive is the approach. Each level of the hierarchy converges at its own spacing, the fastest roads are the sparsest, and a reach that is many lane spacings but only two motorway spacings is close to one hull and far from another. Whether the reach set of a real network at a real distance is near any limit at all, or sits permanently between the limits of its levels, is a question the one-spacing grid cannot answer.