A crossing bends by a law only a conformal chart can show

The quickest route is not the shortest puts a vehicle in water that moves, and its quickest track bends gradually, because the current changes gradually from place to place. A crossing is a chain of decisions re-plans that track as a forecast fails, and the rule that keeps a route near land is pinned at both ends removes the medium and fences the route in instead. None of them has the commonest arrangement a navigator meets, which is a speed that does not vary smoothly at all. It jumps.

A ship working through pack ice makes a few kilometres an hour; the same ship in open water makes twenty-five. The ice edge is a line on the chart, and on either side of it the ship’s speed is, to a first approximation, constant. The quickest route from inside the ice to a point in open water is not the straight line between them, and the way it differs is governed by a law with nothing in it but angles.

Where the bend has to be #

The quickest route is two straight legs, because on each side the speed is constant and a straight line is quickest wherever the speed does not change. The only freedom is where the legs meet the edge, and the argument for where that must be takes three sentences.

Slide the meeting point a short distance δ along the edge. The leg in the ice gets longer by δ times the sine of the angle it makes with the perpendicular, and the leg in the water gets shorter by δ times the sine of its own angle — to first order, which is all that matters for a small slide. So the time changes by $\delta(\sin\theta_1/v_1 - \sin\theta_2/v_2)$, and at the quickest meeting point no slide in either direction can help, which means that bracket is zero:

$$\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}.$$

It is Snell’s law, and nothing about light is needed for it. It is what minimising the sum of two lengths, each divided by its own speed, does to the point where they join.

The figure’s numbers make the trade concrete. The straight line reaches the edge 10.9 kilometres along and spends most of its length in the ice. The quickest route gives up on heading towards the destination at all for its first leg: it runs almost straight at the edge, twelve degrees off the perpendicular, spending 12.3 kilometres in the ice at six kilometres an hour, and then turns hard — sixty degrees off the perpendicular — to cover the remaining 20.1 kilometres at twenty-five. Twenty-four minutes of a three-hour passage come from nothing but where the turn is made.

A law found by the search, not put into it #

The route in that figure is not constructed from the law. Its meeting point is found by minimising the time directly — a search on one number, since the time is a smooth bowl in the meeting point with a single bottom — and the angles are then measured off the result.

That order matters for what the next figure can claim. If the route had been built by solving the law for the meeting point, finding that it obeyed the law would be finding the input again. Built by minimising, the law is an output, and how closely the route obeys it is a measurement of whether the minimisation found the bottom.

The law has only angles in it #

The destinations in that figure are strung out from a few kilometres along the edge to nearly sixty, and the angles in the water run from almost straight out to almost parallel with the edge. None of that spread shows. Every crossing lands on one line through the origin, whose slope is the ratio of the speeds and nothing else.

That is the property that makes the law useful to a navigator rather than merely true. It contains no distance, no position and no coordinate. A planned crossing can be checked without computing anything, by measuring two angles at the point where the track meets the edge and comparing the ratio of their sines with the ratio of the two speeds — with a protractor, on a chart.

On the right chart.

Which charts keep the law #

A protractor measures angles on paper, and a chart is a picture in which ground angles have been turned into paper angles by whatever projection drew it. The law survives the trip onto paper only if both of its angles do.

The three charts that read the law exactly are Mercator, the stereographic and the conformal conic, and they are the three whose only defining property is that they preserve angles at every point. That is not a coincidence to be admired. A conformal chart magnifies every direction at a point by the same factor, so every angle at that point is drawn exactly, and a law made of two angles at one point is drawn exactly with them.

Every other chart magnifies some directions at a point more than others. Measuring instead of naming computes that stretch from each projection’s own derivatives, and it is what the figure uses: the page direction of each leg and of the edge comes from the chart’s own Jacobian at the crossing, and the angles are measured between them on the page. A chart that stretches east–west more than north–south, as the plate carrée does at sixty degrees by a factor of two, pulls every direction towards the east–west line — and it pulls the steep leg in the water and the shallow leg in the ice by different amounts, so the ratio of their sines is no longer the ratio of the speeds.

Read off a plate carrée at sixty degrees north, the crossing implies that the ship makes 10.2 kilometres an hour in the ice. On the Lambert cylindrical equal-area chart, whose east–west stretch at that latitude is four times its north–south, it implies 16.4. On Robinson, measured forty degrees east of its central meridian where its meridians lean steeply, the page angle in the ice comes out at 2.1 degrees instead of twelve, and the crossing implies a ship crawling at one kilometre an hour.

Two charts in the list come close without being right, for different reasons. The Lambert azimuthal equal-area, centred at forty-five degrees north, reads 1.01 because the crossing is only fifteen degrees from its centre, and an equal-area azimuthal chart is still nearly conformal that close in. Mollweide, measured forty degrees east of its central meridian, reads 1.09 by the accident of how its stretch is oriented at that one place, and would read something else a few degrees away. A per cent is small. It is also exactly the size of error a protractor can see, and the law has no tolerance of its own: a crossing drawn on those charts will be judged slightly wrong when it is right.

What conformality buys here, and what it does not #

Conformal does not mean the angles are right is the necessary warning beside this result. A conformal chart preserves the angle between two curves where they meet; it does not preserve the angles of a triangle drawn with a ruler between three distant points, because the ruler’s straight lines are not the ground’s straight lines.

The law of the crossing is the first case, and that is why conformality is exactly what it needs. Both angles are angles at one point — the place where the track meets the edge — between the edge and the track’s own direction there.

What conformality does not supply is straight legs. A navigator drawing the planned crossing on Mercator rules two straight lines on the chart, and a straight line on Mercator is a rhumb line, which leaves the crossing point at a slightly different bearing from the ground’s straight leg. Why Mercator exists is about that difference over an ocean. Over a crossing of a few tens of kilometres it is small: to leading order it is half the convergence of the meridians across the leg, which at sixty degrees north is two hundredths of a degree for the ice leg in the figure and fourteen hundredths for the water leg. Against angles of twelve and sixty degrees, neither is visible to a protractor.

How the error grows with latitude #

Both non-conformal curves start at exactly one on the equator, where each cylinder is locally undistorted, and both climb towards the pole as the east–west stretch grows — as the secant of the latitude on the plate carrée and as its square on the equal-area cylinder. That is why the equal-area cylinder at forty-five degrees reads exactly what the plate carrée reads at sixty: the secant squared of forty-five and the secant of sixty are both two, and a chart’s error in reading this law depends on nothing but how unequal its stretch is.

The third curve is the warning against a correction. Turn the ice edge so that it runs north and south, and the plate carrée’s stretch now pulls the legs away from the edge’s perpendicular instead of towards it, and the reading falls below one. No single factor printed in a chart’s margin can repair the law, because the factor depends on the direction of the edge as well as on the latitude.

Ice edges are features of high latitudes, which is where every one of these curves is steepest. A planner using a plate carrée for a crossing at seventy-five degrees north — which is what a plotted table of latitudes and longitudes becomes by default — is reading the law with an error of more than a factor of two.

The chart the ice is usually on #

The most widely used gridded records of sea ice are laid out on a polar stereographic grid, and that choice is usually explained by what it does for area and for the pole: the grid covers the whole cap without a singular point, and its cells change size slowly and predictably with latitude.

The law of the crossing supplies a reason nobody gives for it. A polar stereographic grid is conformal, so an ice edge traced off one of those grids and a route planned across it can be checked against each other with a protractor at any latitude and for an edge running in any direction. The same edge exported to a latitude–longitude table and plotted as it stands is on a plate carrée, and every angle a planner then measures at seventy degrees north is wrong by the amount the second curve above shows.

The difference is invisible in the picture. Both drawings show the same ragged line and the same planned track; only one of them lets the track be tested against the law the ship will actually follow.

When the quickest route leaves the ice #

The law has a second consequence, and it produces a route that no straight line from start to finish can resemble.

As the angle in the ice grows, the law pushes the angle in the water towards ninety degrees, and at one angle it gets there: the water leg runs along the edge itself. That is the critical angle, $\sin\theta_c = v_1/v_2$, and for six kilometres an hour against twenty-five it is 13.9 degrees.

A route that meets the edge at that angle can run along the edge in the water at full speed and turn back into the ice at the same angle wherever it likes. So between two points that are both in the ice, there are two candidates: straight through the ice, or out at the critical angle, along the edge and back in at the critical angle. Which one is quicker depends only on how far apart the points are.

Where leaving the ice starts to pay #

The straight route’s time grows at the ice’s rate, ten minutes for every kilometre of separation. The route along the edge pays a fixed toll to get out to the water and back — about 155 minutes for points eight kilometres in — and then grows at the water’s rate, 2.4 minutes a kilometre. A steep line and a shallow line with a head start always cross, and they cross at 20.44 kilometres.

The closed form for the crossing separation is short, and the search is there to test it rather than to confirm a formula already believed. Every pair of entry and exit points along the edge was tried on a grid of six hundred positions a side, and the quickest time found agrees with the closed form’s to within two parts in ten thousand at every separation — a bound set by the grid’s own coarseness, not a disagreement.

This is the same arithmetic a refraction survey uses in geophysics, where a signal travelling along a faster layer below the ground overtakes the one travelling directly through the slower layer above it, beyond a crossover distance given by the same expression. There it is used to measure the depth of the layer from where the crossover falls; here the depth is known and the crossover is the planning quantity.

Never nearer than two depths #

The curve says two things a navigator can use without a calculation.

The first is its floor. However fast the open water, the route out and back cannot pay for points closer together than twice their depth inside the edge, because in the limit it is a trip straight out to the edge and straight back, and that trip is two depths long before anything is saved. Points five kilometres inside the ice and eight kilometres apart are never worth taking out to the water, whatever the ship makes there.

The second is how quickly the floor is approached. At a speed ratio of four the crossover is already 2.58 depths, within thirty per cent of the floor. Most of the benefit of fast water is bought by the first few multiples of speed, and a ship that makes eight times its ice speed in open water gains little over one that makes four in deciding whether to leave.

The same angle in a set drawn on roads #

The critical angle has appeared before in these essays, under a different name.

The reach set takes the shape of the roads finds that a reach set on a grid of fast roads over slower ground converges to the convex hull of a road diamond and a ground disc, and that the hull’s straight edges run from the diamond’s corners to points on the disc. Each of those straight edges is a family of journeys that drive along a road and then leave it across the ground — and they leave it at exactly the angle whose sine is the ratio of the ground’s speed to the road’s. The road is the open water, the ground is the ice, and the hull’s tangent is the route along the edge seen as a set rather than as a track.

Where the model stops #

The edge is straight and it does not move. Real ice edges are ragged at every scale and they drift with wind and current, often by kilometres a day. The law holds at every point of a curved edge with the perpendicular to the edge’s local tangent, but the single crossing point of the straight case can become several.

The speeds are constants. A ship’s speed in ice depends on the ice’s concentration and thickness, which vary continuously inside the edge, so the ice side is really a smoothly varying medium like the flows the quickest route is not the shortest handles, with a jump only at the edge itself.

The turn is free. The route bends through forty-eight degrees at the edge and a ship cannot; the shortest route a vehicle can fly prices a bounded turning radius. For a turn of that size on a radius of a kilometre, cutting the corner with an arc costs 2r tan(θ/2) − rθ, about fifty metres, against a crossing of more than thirty kilometres.

And the ground is a plane. Every distance here is tens of kilometres, over which the difference between a plane and the sphere is far below any angle the law is read to.

Still open: an edge that curves #

A straight edge gives one quickest crossing between any two points, and the critical-angle route gives at most one alternative. A curved edge need not.

Where an ice edge bulges out into the water, the quickest routes from one point in the ice to a spread of destinations in the water fan out from the bulge at angles the law sets locally, and routes leaving neighbouring stretches of a convex edge can cross. Where two of them cross, a destination has two quickest crossings of exactly equal time — the same failure of uniqueness that where the shortest route stops being the only one finds for geodesics on an ellipsoid. Whether a bulge in an ice edge produces such a set, how large it is, and whether a chart could show it, is a question a straight edge cannot ask.