Chebyshev's map is the best at its worst, and not on average

The nodes were evenly spaced made the fitted map settle: with its collocation nodes crowded towards the corners, a conformal map of a spherical square holds its boundary scale constant to seven parts in a hundred thousand, and stays there as the nodes are refined. It ended on a question every construction here had assumed the answer to. Every construction imposed its condition on the boundary, because that is where Chebyshev’s criterion says it belongs, and the interior was left to look after itself.

Whether it does is a question with a precise form. Chebyshev’s criterion is a theorem about the worst case: among conformal maps of a region, the one whose scale is constant on the boundary has the smallest ratio between its largest and smallest scale. It says nothing about how the scale is distributed over the area in between, and a reader of the map lives in the area, not on its edge.

A condition on the boundary is a statement about the worst case #

The reason Chebyshev’s criterion is about the boundary at all is a property of conformal maps. For a conformal map of the sphere, the logarithm of the scale is the sum of two parts: a fixed function of distance from the region’s centre, which comes from the sphere’s own curvature, and a harmonic function, which is the whole of the freedom the cartographer has. A harmonic function takes its largest and smallest values on the edge of any region it is defined on, and that is why the extremes of scale are decided at the boundary.

So the way to make the range as small as possible is to spend the freedom making the largest scale as low as it can go everywhere on the edge at once, which means making it the same everywhere on the edge. The smallest scale then falls wherever the fixed part is least — at the centre — and the ratio of the two is the least any conformal map allows. That is the theorem, and solving for the map instead of choosing it turned it into a fit.

What the construction also does, by the same logic, is decide the interior completely. Not every distortion can be asked for is the essay about why: a conformal map’s scale satisfies an elliptic equation, and an elliptic equation’s solution is fixed by its boundary values. Once the scale is constant on the edge, there is no freedom left inside. The interior is whatever that boundary forces it to be.

The same series, asked a different question #

The comparison worth making is with the map that spends the same freedom on the interior instead.

Chebyshev’s map is fitted by asking that the logarithm of the scale equal one constant at every boundary node. The same unknowns — the coefficients of the series and the constant — can be asked to make the logarithm of the scale as close to a constant as possible over the whole area, in the mean square, with each part of the region counted by its area on the sphere. That is also a linear least-squares fit. It differs from Chebyshev’s only in where the condition is imposed: at the interior, weighted by area, rather than on the edge.

For a conformal map this is not an arbitrary second criterion. A conformal map stretches every direction equally, so its two principal scales are one number, and Kavrayskiy’s index of distortion — half the sum of the squared logarithms of the two principal scales — is exactly the squared logarithm of the scale. The mean-square map is the conformal map of a region with the least Kavrayskiy index, and to the same order the least Airy index. The average was a choice of norm found that ranking named projections by a mean and by a worst case gives different orders; this is the same pair of norms asked not which existing map is best but what map each one builds.

The cap, where the question makes no difference #

On a spherical cap the two questions must have the same answer, and it is worth seeing why before seeing where they part.

The fixed part of the log scale depends only on distance from the centre, and so does the cap. Every non-constant harmonic function, averaged round a circle about the centre, is zero, so none of them can reduce the mean-square misfit against a function that is constant round every circle. The best harmonic part is a constant, the map is the stereographic projection centred on the cap, and that is also Chebyshev’s answer — constant on the rim by symmetry.

Computed, the two fitted maps of a 25° cap agree in the logarithm of their scale to $1.7 \times 10^{-15}$ at every point sampled. They are one map.

That result was not free. The first version of the mean-square fit sampled the interior on a rectangular grid, which is not symmetric about the cap’s centre, and a twelve-term series used the asymmetry: it returned a map whose range was five and a half per cent wider than the stereographic’s while matching its mean square to four figures. Sampling in rings about the centre, each point weighted by its exact area on the sphere, removed it. The same instrument is used for every region below, and a mean-square fit that did not reproduce the stereographic on a cap would not be trusted anywhere else.

On a square, two different maps #

A spherical square has corners, and the fixed part of the log scale is not constant along its edge: the corners are further from the centre than the middles of the edges. Chebyshev’s map has to cancel that difference exactly, all the way round.

Along the ray to a corner, Chebyshev’s map rises from 1.54 per cent below its mean at the centre to 1.44 per cent above it at the corner. Along the ray to the middle of an edge it rises to exactly the same 1.44 per cent, over a shorter distance and so more steeply. That is what constant on the boundary means in the interior: the map works hardest along the short rays, to bring the middles of the edges up to the level the corners are at.

The mean-square map does not do that. It lets the four corners rise to 2.05 per cent above its mean and holds the middles of the edges at 0.11 per cent below it. The corners are small and the stretches of edge between them are long, so the area is served better by giving up the corners than by raising everything else to meet them.

That is visible in the hero figure as the difference between lines of equal scale that follow the boundary and lines that cross it. On the left the boundary is itself a contour, and every contour inside is a rounded square nested within it. On the right the contours are closer to circles, and the boundary cuts across them, with the four highest ones confined to small patches at the corners.

Where the area goes #

The two maps’ scales can be compared not just at their extremes but as distributions over the square’s area.

Under Chebyshev’s map, 68 per cent of the square departs from the mean scale by more than half a per cent. Under the mean-square map, 56 per cent does. At one and a half per cent the order reverses: 1.7 per cent of the area under Chebyshev’s map and 4.3 under the other. Beyond 1.54 per cent — its departure at the centre — Chebyshev’s map has no area left at all, and the mean-square map still has its corners, out to 2.05.

So the answer to the question the previous essay left is specific. A map whose boundary scale is constant has an interior, and the interior is not bad — but it is not the interior anybody would design for the interior’s sake. It carries more of its area at moderate departures than it needs to, because it has been bent to guarantee something about a line of zero area.

A path between them #

The two maps are the ends of a trade rather than two unrelated answers, and that can be shown exactly.

The logarithm of a map’s scale depends linearly on the series’ coefficients, so the maps on the straight line between the two sets of coefficients have log scales that are the corresponding blends. Along that line both criteria can be computed exactly. The root-mean-square falls steadily from Chebyshev’s end to the other, from 0.870 to 0.804 per cent, and the range rises steadily, from 2.98 to 3.47. No step improves both.

That is what makes this a trade rather than a refinement. If Chebyshev’s map were a slightly worse version of the mean-square map, or the reverse, some step along the way would improve both numbers at once. None does, so each criterion is paying for something the other does not value. A map maker choosing between them is not choosing the better map. They are choosing whether the edge of the region or the bulk of it is the thing to protect.

Four regions, and what each loses #

The square is one shape. The trade depends on shape, and three more regions of the same size show how.

On the cap both losses are zero, as they must be. On the square Chebyshev’s map is 8.3 per cent worse in root-mean-square and the mean-square map 16.5 per cent wider in range. On an ellipse the two losses are nearer each other, 10.7 and 9.2 per cent. On a sliver — an ellipse five times as long as it is wide — they separate the other way: Chebyshev’s map is 19.8 per cent worse in root-mean-square, and the mean-square map gives up only 2.6 per cent of range.

The sliver’s numbers are the most practical ones here, because many of the regions a single conformal grid or chart is asked to cover — a coastal state, a corridor, a long island chain — are long and thin. On a region like that, holding the boundary constant costs a fifth of the achievable mean-square accuracy to protect a range that the alternative gives up almost none of. Chebyshev’s guarantee is cheapest where the region is most nearly round and dearest where it is least.

The pooled score abandons a region found the same two objectives choosing different projections for a set of regions, where the worst case protects the region the mean would sacrifice. Here the regions are the parts of one region, and the part the mean gives up is its corners.

Which question a purpose is asking #

The two maps are both optimal, so the choice between them is not a matter of accuracy. It is a matter of what the map is for, and the difference can be put in the terms a user of a grid actually states.

A tolerance that must hold everywhere is Chebyshev’s question. How small is flat enough is written that way — the size of a patch at which the error of treating it as flat stays inside a stated bound at every point — and how many sheets an atlas needs inverts exactly this bound into a sheet radius and a sheet count. On the square, Chebyshev’s map keeps every point within 1.49 per cent of a single scale. The mean-square map, with its overall scale set to do as well as it can by the same test, keeps 96.5 per cent of the area within that tolerance and needs 1.74 per cent to cover the rest. A guarantee that fails on three and a half per cent of a territory is not a guarantee, and for a tolerance written into a contract Chebyshev’s map is the only one of the two that meets it.

A tolerance that describes most of the area is the other question. Tighten the bound to half Chebyshev’s, 0.74 per cent, and ask how much of the square each map keeps inside it: Chebyshev’s map keeps 53 per cent and the mean-square map 61. The ellipse gives 52 and 65, and the sliver 52 and 75. A map that is going to be measured on — distances read off it at places spread over the territory, areas totted up field by field — is being judged on that second kind of number, and on the sliver the mean-square map puts nearly a quarter more of the territory inside the tighter bound while giving up a quarter of one per cent of area at the looser one.

So the purpose sorts itself by a single word. If the requirement says every, the boundary condition is the right one and its price is the interior’s mean. If the requirement says typically, the mean-square map is the right one and its price is the corners. The criterion that has carried Chebyshev’s name for a hundred and seventy years answers the first, and a map maker whose requirement is the second has been using it by default.

Shape, not size #

The last measurement asks whether any of that depends on how big the region is.

From a square of 5° circumradius to one of 40°, Chebyshev’s own root-mean-square departure grows seventy-four times, from 0.033 per cent to 2.43. The trade between the two maps hardly moves: Chebyshev’s root-mean-square excess goes from 9.5 per cent to 6.5, and the mean-square map’s range excess from 17.0 to 15.5.

That is what a trade set by the square’s corners would do. A small spherical square is very nearly a flat square, and the flat square has the same corners at any size; the curvature adds a little on top as the square grows, and the curves drift slowly rather than level off. The price of holding the boundary constant is a property of the region’s shape, and it can be computed from a territory’s outline at whatever size is convenient.

What each number was checked against #

Every measurement is taken where neither fit was imposed. Both maps are scored on a ring sample finer than either fit used, the habit a condition imposed at points is not a condition showed a residual needs.

The two fits are the same code asked two questions. The step that turns a solved set of coefficients into a map was separated out and is shared by both, so a difference between the maps cannot be a difference between two implementations; Chebyshev’s map on a 20° cap still reproduces the stereographic spread to fifteen figures after the change.

On a cap the two maps must be one. They agree to $1.7 \times 10^{-15}$ in the logarithm of the scale, which is the refusal the whole comparison rests on: a region with nothing for the two criteria to disagree about must produce no disagreement.

On every other region each must lose on the other’s criterion, by more than one per cent, and only Chebyshev’s map may hold its boundary scale constant: on the square its boundary varies by 0.012 per cent against the mean-square map’s 2.2.

And the path between them must be monotone in both criteria at every step, which is the check that the trade is a trade.

What the comparison leaves out #

Twelve terms, one series. Both maps are the best of their kind within a truncated series. A longer series lowers both criteria, and nothing here says the ratio between them survives unchanged.

Two criteria from one family of norms. The mean square and the worst case are the two ends the collection’s published indices use. A criterion counting only the area within a stated tolerance is a third, and it is neither of these.

Four shapes. The regions are a cap, a square, an ellipse and a sliver, all centred and symmetric. A real territory’s outline has corners of different angles and no symmetry, and the trade would be read off that outline rather than off these.

Still open: the map that keeps the most area within a tolerance #

A surveyor does not usually want the smallest worst case or the smallest mean square. They want as much of the territory as possible within a stated tolerance of true scale — within one part in a thousand, say — and they are indifferent to how far outside it the rest goes.

That criterion is neither of the two here. It counts area, like the mean square, and it ignores everything beyond a threshold, like neither. The tail figure shows the two maps crossing, which says the answer lies somewhere else again. Every projection minimises something records the objectives cartographers have written down, and this one has the unusual property of not being convex: the area within a tolerance can have several local optima, and a fit started from Chebyshev’s map and one started from the mean-square map need not arrive at the same place. Which conformal map of a region keeps the most area within a tolerance, whether it looks like either of these, and how much area it gains over them, are questions two convex criteria cannot ask.