A cap on the worst departure keeps the thin regions' gain and gives back the square's

Three maps make a plane, and the band’s best map lies outside their triangle took three solved conformal maps of a region — the p = 1 map, which minimises the mean departure from true scale, the least worst map, and the mean-square map — and used the fact that a conformal map’s ln k is linear in its coefficients. Every blend of the three is a conformal map of the same series, so three solved maps are a whole plane of them, and the plane can be searched for the map that keeps the most of a region inside a surveyor’s band ±τ — the criterion the map that keeps the most ground inside a tolerance introduced, and that two maps and the line between them do the work of every exponent first searched along a line. The best map in the plane kept up to twenty points more than the best of twelve exponents. It also let the edges of the region go: on the square, at a band of half the least worst departure, the best map’s own worst departure was 3.8 times the least worst.

A surveyor who accepts the band but not unlimited error outside it is asking a question with two clauses — the most area inside ±τ, among maps whose worst departure stays below a cap. That question has a shape in the plane, because the worst departure is the largest of a set of absolute values of functions linear in the two coordinates, and so it is convex. The maps under any cap form one convex region, and the search is over that region instead of the whole plane.

The earlier essay ended with three questions about it: whether the area kept falls smoothly as the cap tightens or collapses as soon as the cap excludes the far-out maps, whether the best capped map moves back into the triangle or slides along the cap’s edge, and whether a cap of twice the least worst leaves most of the gain in place. Each has a measured answer, and the first depends on the region’s shape so strongly that it answers the third as well.

A cap is a long band of maps, and the best map rides its edge #

The plane is the same one the earlier essay drew. Write $\ln k_1$, $\ln k_\infty$ and $\ln k_2$ for the three maps’ fields; every map in it has the field

$$\ln k(s, u) = \ln k_1 + s\,(\ln k_\infty - \ln k_1) + u\,(\ln k_2 - \ln k_1),$$

and its worst departure is the largest $|\ln k|$ over the region’s samples. Measured as a multiple of the least worst departure w*, every cap from 1 to 5 picks out a convex region of the plane. On the square those regions are not discs round the least worst corner, as a reader might picture a tolerance. They are long, narrow, nested bands running diagonally through it, because the three fields nearly share one direction along which the worst departure changes slowly: moving along the band trades error at one part of the boundary for error at another, and moving across it adds error everywhere.

Searched over every cap from 1 to 5 in steps of a twentieth to a half, at seven bands from 0.3 to 0.9 of w*, on thirteen regions — six ellipses from 1 : 0.8 to 1 : 0.12, a spherical triangle, a square and a hexagon, and four regions with a bay cut into one side, all 25° in their long radius — the cap binds in 1,045 of the combinations, meaning the best map under it keeps less than the uncapped plane’s best. In 791 of those the capped best lies on the cap’s own edge, its worst departure within half a per cent of the cap. In 151 it lies inside or on the triangle of the three maps. So the answer to the second question is the second of its two options. Tightening the cap does not draw the best map back between the three solved maps; it slides it along the edge of what the cap allows.

The slide is not one continuous path. On the square at half the least worst departure the best map under a cap of 1.05 to 1.5 sits on the mean-square map’s side of the plane, at u between 0.7 and 1.6. Between 1.5 and 1.8 it crosses to the other side — (0.35, 0.20) at 1.6, (0.50, −0.24) at 1.8 — and from there it walks steadily outward: (0.50, −0.36) at a cap of 2, (0.55, −1.63) at 3 — until at 3.82 it reaches the uncapped best at (0.72, −2.81). A tight cap and a loose one are answered by maps in different directions from the least worst corner, and a surveyor who loosens the cap a little at the wrong place moves the answer across the plane. It is the same instability a criterion worth using is one whose answer is not unique found in the band criterion itself, now showing up in the cap that was meant to tame it.

The exponents already sit on the frontier, and the plane extends it #

The cap turns the earlier essay’s one-number answer into a curve. For each cap there is a most-area map, and plotted against the cap those maps trace the frontier of area against worst departure — every map off it is beaten on both counts by a map on it. It is a Pareto front, of the kind that essay drew for whole projections, drawn here inside one family of conformal maps of one region.

The twelve exponents fall on it or close to it. Each is a map with a worst departure and an area kept, so each is a dot on the same axes. On the thin ellipse the members from p = 1 to p = 3 sit within four per cent of the frontier, and p = 1 on it to the width of a line: at half the least worst it keeps 65.3 per cent at a worst of 1.58, and the least worst departure at which any map in the plane keeps as much is 1.56. The members above p = 3, crowded near the least worst corner, sit further inside. On the square the exponents sit a little inside: p = 1 keeps 66.4 per cent at 2.20, and the plane keeps as much at 2.04, 7.5 per cent lower. Over all thirteen regions at that band, the plane keeps p = 1’s area at a worst departure between 0.3 per cent lower — on the two thinnest ellipses — and 10.4 per cent lower, on the roundest. The exponent family was never wasting much. What it lacked was reach: it stops at p = 1, and the frontier carries on — in the direction one exponent names one map, and at tight bands a better one followed below p = 1 until its answer stopped being unique.

The other thing the frontier shows is the one a single number could not. On the square it rises steeply for the first fifth of the least worst departure and then bends over, a curve of diminishing returns. On the ellipse it jumps once at the least worst corner, rises gently, and then steepens, running almost straight up between 1.6 and 1.9 before flattening at the uncapped best. A frontier that steepens has a stretch that bulges inward, below the straight line joining its ends, and that stretch matters for a reason the next section makes concrete.

On a thin region the gain is a pocket, admitted all at once #

The first of the three questions has two answers, and the region decides which. On a thin ellipse the area kept does collapse when the cap excludes the far-out maps, but the collapse is not toward the least worst map. It is a step down off a small pocket.

On the ellipse of 1 : 0.45 at 0.3 of the least worst departure, the band’s best map keeps 59.8 per cent of the region and sits at (0.07, −2.28) with a worst departure of 2.11. The maps near it that keep within three points of that are a small enclosed patch of the plane, about half a unit wide and a unit long. Every cap that reaches 2.11 admits the whole patch at once and keeps 59.8 per cent; a cap of 2 excludes it and keeps 53.7, the best it can do being a map on its own edge farther out at (0.60, −3.09); a cap of 1.9 keeps 49.5, 1.7 keeps 44.6 and 1.5 keeps 41.3, a tenth of a point under the best exponent. Between caps of 1.5 and 2.11 the region gains eighteen and a half points, and more than six of them between 2 and 2.11.

On the square, the same reading gives a ramp. At half the least worst departure the square keeps 61.3 per cent under a cap of 1.5, 65.8 under 2, 69.1 under 2.5, 72.1 under 3 and 75.8 under 4. Past the first tenth of the least worst departure, where it climbs five and a half points at once, no quarter of the least worst adds more than two and a half points, and the best map moves steadily outward as the cap widens. There is no pocket to fall off: the map that keeps the most at each cap is simply the farthest one the cap allows along a direction that keeps paying.

How much cap the gain needs, region by region #

Two caps describe each region. The first is the smallest cap under which the plane keeps as much as the best exponent keeps uncapped — the price of matching the family. The second is the smallest under which it keeps nine tenths of what the uncapped plane gains over the family — the price of taking nearly all the gain.

On the thin ellipses the two are close together, and closer the thinner the region. At 0.3 of the least worst departure they are 0.55 apart on the ellipse of 1 : 0.45, 0.38 on 1 : 0.3, 0.22 on 1 : 0.2 and 0.20 on 1 : 0.12. On those four, nine tenths of a gain of ten to eighteen points arrives within a fifth to a little over a half of the least worst departure past the point where the plane first equals the family. On the triangle, the square and the hexagon the two caps are 1.25, 1.17 and 0.68 apart, and on the deep bay 2.24: the gain is spread over a range of caps several times as wide.

That is the answer to the third question, and it is not one number. A cap of twice the least worst leaves the whole gain in place on the ellipses of 1 : 0.3, 1 : 0.2 and 1 : 0.12 at every band, and on 1 : 0.45 at every band but the two tightest, where it keeps 67 and 98 per cent of it, because their pockets lie at worst departures of 1.1 to 2.1. On the square, the triangle, the hexagon and the four bays, at bands of 0.3 and 0.5 of the least worst, it keeps at most 28 per cent of the gain, and on five of the seven less than the exponents had to begin with: at half the least worst the square under a cap of 2 keeps 65.8 per cent, below the exponents’ uncapped 66.4. Over all thirteen regions and seven bands, a cap of 2 matches or beats the uncapped best exponent on 69 pairs of 91, a cap of 2.5 on 87 and a cap of 3 on 90. The surveyor’s useful rule is about the region’s shape before it is about the cap: a thin region can be capped at twice the least worst for nothing, and a compact one cannot be capped below three without giving most of the plane’s gain back.

Twice the least worst already rules out the best exponent on five regions #

The comparison in the last paragraph flatters the exponents, because it compares a capped plane with an uncapped family. A cap applies to every map, and it applies to the exponent maps too. The p = 1 map, which is the best exponent at tight bands on every region here, has its own worst departure: 1.44 to 1.58 of the least worst on the six ellipses, 1.90 on the triangle, 2.20 on the square, 2.32 on the hexagon, and 2.54, 2.61, 1.99 and 2.84 on the four bays. A cap of twice the least worst excludes it on five of the thirteen regions — the square, the hexagon and three of the bays — and a surveyor who imposes that cap and then reaches for the exponent family finds that the member it would have chosen is not available.

The exponents are ordered by their worst departure, which falls steadily from p = 1 to p = 32, so a cap cuts the family at some exponent and admits everything above it. The best admitted exponent is then usually the smallest one the cap allows. On the square at half the least worst departure that is p = 1.25, keeping 63.3 per cent under a cap of 2 where the plane keeps 65.8; on the deep bay it is p = 2, keeping 65.4 where the plane keeps 68.5.

The ticks on the figure say the same thing from the other side. They mark the least worst departure at which some map in the plane keeps as much as p = 1 does, and on every region they sit just to the left of p = 1’s own dot: the closest are the thin ellipses, where the two coincide to a hundredth, and the farthest the ellipse of 1 : 0.8, where the plane needs 0.90 of p = 1’s worst. No cap below that tick can be satisfied by anything that keeps p = 1’s area, in the plane or in the family.

Under the same cap, the plane beats the exponents it admits #

The fair comparison holds the cap fixed for both sides: the plane’s best map under the cap against the best exponent whose own worst departure the same cap admits. Measured that way the plane is better at every cap, and by the most where the cap is tightest or the pocket has just been admitted.

The tight end is where the exponents are weakest. A cap of 1.1 admits only the exponents from about p = 6 upward, all of which are close to the least worst map and keep little inside a tight band. The plane under the same cap can go along the cap’s long band to maps far from the least worst corner that are just as tight in their worst departure and much better in the band: on the triangle at 0.3 of the least worst, 45.6 per cent against the admitted exponents’ 31.9, a gain of 13.7 points; on the square under a cap of 1.3, 9.2 points. The loose end is where the thin regions’ pockets are admitted, and the ellipse of 1 : 0.45 gains 18.4 points over the family from a cap of 2.25 on, the whole of its uncapped gain.

Over every cap from 1 to 5, every band and every region, the median gain is 1.9 points. In a few places the plane falls short of an admitted exponent, by at most 0.86 points, because the exponents between p = 1.5 and p = 8 lie slightly off the plane — up to 0.13 of their own departure on the thinnest ellipse, as the earlier essay measured — and a cap that admits one of them does not admit a map in the plane that reproduces it exactly.

A cap can choose maps a weighting cannot #

The steepening frontier of the thin ellipse has a consequence that is about criteria rather than about maps. A natural alternative to a cap is a weighting: score each map by its area kept minus some multiple of its worst departure, and take the best score. A weighted score of two quantities is maximised only at corners of the frontier’s upper hull — the dashed line on the frontier figure — because a straight line of constant score touches the frontier from above. Every map on the frontier that lies strictly below that hull is one no weighting will ever select.

On the square at half the least worst departure about half the frontier’s range of caps lies below its hull, by at most 1.5 points: a weighting misses those maps but loses little by doing so. On the ellipse of 1 : 0.45 four fifths of the range lies below it, by up to 3.3 points, and on the thin ellipses at 0.3 of the least worst the share is 90 to 95 per cent, by up to seven points. A surveyor who prices worst departure against area by a single exchange rate will be offered either a map near the exponents or the pocket’s map at 2.11, and never anything between — never, for instance, the map that keeps 49.5 per cent under a cap of 1.9, eight points more than the exponents, for a worst departure a tenth smaller than the pocket’s.

That is the same shape which projection a weighting can make best found among whole projections, where four of seven undominated world maps sat in dents of the front and no weighting of angle against area could put them first. Here the dents are inside one family of conformal maps of one region, and they are most of its frontier wherever the region is thin. A cap reaches every point of a frontier; a weighting reaches only its hull. A second clause decides the half that was already decided used a second clause to break ties; a cap is a second clause that decides which part of the frontier is on offer at all.

The map called least worst was not the least #

Every cap here is a multiple of the least worst departure, and the least worst departure has been, throughout this comparison and the two before it, the worst departure of the exponent family’s p = 32 member. That map approaches the minimax map as the exponent grows, which is why it was used; it has not reached it. Searched for the smallest worst departure anywhere in the plane, on the same samples, every region has a map below p = 32’s: by 0.2 per cent on the hexagon, 0.5 on the square, 0.3 to 2.4 on the ellipses, and 4.0 to 5.9 per cent on the four regions with a bay.

So a cap of 1 does not admit a single map. It admits a small region of maps round a floor below it, and on the bays a region of some size: on the ellipse of 1 : 0.3 with a bay the floor is 0.94 of the least worst, at (1.18, −0.30). The bays are where the worst point is not on the grid of any simple sampling and where a power of 32 still weights the second-worst samples heavily enough to pull the map away from the minimax.

None of the conclusions above depends on the choice, because every cap is stated relative to the same number on every region and every comparison is made at a fixed cap. What changes is how a reader translates a cap of 2 into a statement about the map: it is twice p = 32’s worst, and on a bay that is 2.1 times the plane’s own floor, not twice it. The true minimax over all twenty-five coefficients would be lower again, and Chebyshev’s map is the best at its worst, and not on average is the measurement of that map rather than of its p = 32 approximation.

Monotone, fair, convex and fine enough: the four checks #

Loosening the cap must never lose area. The cap is a filter, so every map admitted by a cap is admitted by every looser one. At every band on every region the area kept must not fall from one cap to the next by more than three tenths of a point, the resolution of the refinement; it does not fall at all.

The capped plane must hold what the admitted exponents hold. Every exponent lies close to the plane, so under any cap the plane’s best must keep within a point and a half of the best admitted exponent. The largest shortfall anywhere is 0.86 points.

The region under a cap must be convex. Four thousand pairs of maps under a cap of twice the least worst on the square, and the midpoint of every pair, must also lie under the cap. Every one does.

The grid must not decide the uncapped answer. The uncapped best is searched on a grid of 0.04 over the whole ±8 window, twice as fine as the earlier essay’s, and refined twice to a five-hundredth. Against that essay’s coarser search the two agree to within 0.44 points at every band on every region, inside the half point the earlier control allowed.

The floor must lie under p = 32. The plane contains the p = 32 map, at (1, 0), so its smallest worst departure cannot exceed p = 32’s; each region’s floor is searched from a grid and refined three times, to a two-thousandth.

Three maps, a sampled worst and synthetic regions #

The same three maps. Every capped search is over the plane of the p = 1, p = 32 and mean-square maps. The earlier essay found that a search over all twenty-five coefficients adds about a point on most regions and seven on the hexagon; under a cap, the full search has not been run.

A sampled worst. The worst departure is the largest over forty rings and a hundred and sixty azimuths. A map that sits on a cap’s edge has its worst at a sample, and the true worst between samples can be slightly larger; on a region with a bay that difference is largest where the boundary turns.

One family of regions. Thirteen regions at one size, and bands from 0.3 to 0.9. Whether real coastlines carry pockets the way thin ellipses do, or ramps the way polygons do, is not something synthetic regions can say.

History. Bounding one objective and optimising the other is the ε-constraint method of multi-objective optimisation, usually traced to Yacov Haimes, Leon Lasdon and David Wismer in 1971, and its advantage over a weighted sum — that it reaches the unsupported parts of a front — is standard. What is measured here is how much of this particular front is unsupported, and where.

Still open: whether the cap’s price belongs to the plane or to the problem #

Every capped answer above is the best map in a plane of three, and three maps chosen for a different purpose. The pocket on the thin ellipse, the long bands of admitted maps, and the crossing from one side of the plane to the other as the cap passes 1.5 on the square are properties of that plane as much as of the region.

The test is the one the earlier essay ran without a cap: start the search over all twenty-five coefficients from the capped plane’s best, and keep every step it takes under the cap. If the full search under a cap of twice the least worst recovers on the square the gain the plane gave back, then the ramp was the plane’s limitation and a compact region can be capped as cheaply as a thin one. If it does not, the price is the region’s, and the frontier drawn here is close to the true one. Whether the capped full search also leaves the pockets where they are — and whether a minimax map solved exactly, rather than approached at p = 32, would move the caps’ bands enough to change which regions pay — are questions a search confined to the plane cannot ask.