Two maps and the line between them do the work of every exponent

The band names the exponent for a compact region, and not for a thin one asked which member of a family of conformal maps serves a surveyor’s tolerance band best. The family minimises the mean of |ln k| raised to a power p: at p = 1 the mean departure from true scale, at large p the least worst case. Read against the region’s own least worst departure w*, a band on a compact region is served by p = 1 up to about nine tenths of w* and by the least worst case above it. A thin region passes through every exponent in between, and a rule that switches from one end to the other gives up five to seven points of area there, wherever the switch is put.

It ended by asking whether the two ends could predict the middle — whether a surveyor who solves the mean map and the least worst map, which the band needs anyway to be read against w*, could read off the answer without solving the rest of the family. The question as it stood was about distributions: whether the members’ shares of area at each departure interpolate smoothly enough for two ends to predict the others.

There is a more direct version of the question, and it has a clean answer. The two ends do not need to predict the members at all. Between them lies a line of maps that can be searched directly, and that line serves every band about as well as the whole family does.

A map between two maps #

The conformal maps here are all built the same way, as the earlier essays built them: a series of twelve complex terms whose real part, added to a fixed term the sphere supplies, is ln k at every point of the region. The fixed term is the same for every map. The series is where the maps differ, and its real part is linear in the series’ coefficients.

That has a consequence the earlier measurements never used. Take the coefficients of the map that minimises the mean departure and those of the least worst map, and blend them: a fraction 1 − t of the first and t of the second, the same t for every coefficient. The result is a series like any other, so it is a conformal map, and its ln k at every point is exactly 1 − t times the first map’s plus t times the second’s. Every map on that line, from t = 0 to t = 1, is known as soon as its two ends are, with no further fitting.

So a surveyor holding the two ends can ask a question the exponent family never asked directly: of all the maps on the line, which keeps the most of the region inside this band? That is a search over one number, done by evaluating the blended field at the region’s samples and counting the area inside the band. Here it is done at 201 evenly spaced positions along the line, for each band from 0.3 to 1.0 of w*, on the ten regions the earlier essay measured — a cap, six ellipses from 1 : 0.8 to 1 : 0.12, and a spherical triangle, square and hexagon, all 25° in their long radius — and on four more with a bay cut into one side.

The line keeps what the family keeps #

On every compact region the line comes within 0.15 points of the best of twelve exponents at every band, and on five of the seven it never trails by as much as a hundredth of a point. On the square the largest shortfall is below a hundredth of a point. What the two-exponent rule gave up on the same regions — 1.2 points on the square, 1.9 on the triangle, 3.1 and 3.8 on the ellipses of 1 : 0.6 and 1 : 0.45 — the line gives up none of.

On the thin ellipses the line does fall short, and by more: 0.84 points at most on the ellipse of 1 : 0.3, 0.96 on 1 : 0.2 and 1.47 on the thinnest. Those are the regions where an intermediate exponent is best over a stretch four tenths wide, and where the rule gave up 5.6, 5.7 and 6.6 points. The line recovers more than three quarters of what the rule lost on each of them.

Two of the rule’s losses in the earlier comparison were larger than these — 2.3 points on the cap and 2.0 on the ellipse of 1 : 0.8 — and both came from a band of one twentieth of w*, narrower than the three tenths that comparison said it started from, where the shares are a few per cent and a handful of samples moves them. Over the bands both comparisons state they cover, from 0.3 upward, the rule gives up 1.3 and 1.75 on those two regions, and every other figure agrees with the earlier one.

The cost of the search is the other half of the result. Solving the family took twelve reweighted fits per region, each continued from its neighbour so that it starts near its answer; the line takes two, and each of the 201 points along it is a weighted sum of two stored fields. A surveyor who is told a band needs the two ends anyway, to know w*, and the line adds nothing to that but arithmetic.

On a square, a map no exponent draws #

On the square the line does better than the family. At 0.85 of the least worst departure the best of the twelve exponents is still p = 1, keeping 88.6 per cent of the square; the map 0.23 of the way along the line keeps 90.3. At 0.9 the best exponent keeps 91.7 and the line 92.8. The line’s advantage on the square reaches 1.67 points, and on every region it is somewhere between 0.05 and 1.67.

The contours show where it comes from. The mean map keeps the middle of the square and loses the four corners, where its departure runs to more than twice w*. The least worst map loses a disc in the middle and a thin strip along the whole edge, because it holds its departure near w* everywhere on the boundary and the band is narrower than that. The map on the line between them gives up a small disc in the middle and keeps more of each corner than the mean map does, and at this band the corners gained are worth more than the disc lost.

That the line can beat the family is less surprising than it looks. Both are one-parameter sets of maps, but they are chosen differently. The exponent family is fixed before any band is named: each member minimises its own average, and the band is only consulted afterwards, to pick a member. The line’s position is chosen by the band itself — it is a search on the very quantity the surveyor asked for, over a set of candidates that happens to run between the two maps that matter most. The map that keeps the most ground inside a tolerance made the same search over every map at once, and a criterion worth using is one whose answer is not unique found it had forty-two answers; a search over one line has one answer at each band, and it sits between two maps the surveyor has already understood.

The members are not on the line #

The obvious reading of the result is that the family’s members lie on the line — that the exponents interpolate between the two ends and the line simply reproduces them. The figure rules it out. Project each member’s ln k onto the line and measure what is left over: at p = 2, the mean square, four tenths of the map’s departure from the mean map is in directions the line does not contain on the square, and nearly two thirds on the ellipse of 1 : 0.2. The members approach the line only near the large-p end, where they are nearly the least worst map already, and on the thin ellipses not even there: at p = 8 two fifths of the departure is still off the line.

So the line and the family are two different routes from the mean map to the least worst map, and on a compact region the line’s route passes through maps at least as good for every band. On a thin region the family’s route bends away through maps the line cannot reach, and a few of those are better — by up to a point and a half on the thinnest ellipse, and only between 0.6 and 0.9 of w*. That is where the earlier comparison found the thin regions’ least worst map holding much of its area at small departures, along the long axis, and it is where the best maps are ones that treat the long axis and the two tips differently from either end.

A concave coast does not break it #

The earlier essay’s regions were all convex, and it named a concave coast as the case where neither end’s behaviour is simple. Four regions with a bay test it. Each is star-shaped from its centre, which the sampling needs, and each has a Gaussian bite taken out of one side: from a cap, from ellipses of 1 : 0.6 and 1 : 0.3, and a deep narrow bay in an ellipse of 1 : 0.8.

A bay is hard on the two-exponent rule. On the ellipse of 1 : 0.6 with a bay the rule gives up 5.1 points, where the same ellipse without one cost it 3.1, and on the 1 : 0.3 ellipse with a wide bay it gives up 7.0, the most of any region measured. Even the cap, where every exponent drew the same map, stops being degenerate once it has a bay: the family’s members differ, an intermediate exponent is best from 0.85 of w*, and the rule gives up 3.1.

The line barely notices. Its largest shortfall on the four regions with a bay is 0.76 points, on the cap, and on the others 0.36, 0.26 and 0.03. On the ellipse of 1 : 0.3 with a bay it gives up a quarter of a point where the rule gave up seven. A concave coast makes the family’s middle matter more, and the line still reaches nearly everything the middle offers.

Where on the line the band puts the map #

What the surveyor reads off the line is a position t, and it behaves as the earlier essay’s exponent did, without the jump. On a compact region the best map is the mean map, t = 0, until the band is between seven and eight and a half tenths of w*, and then it moves quickly along the line, reaching the least worst map as the band reaches w*. On a thin region it leaves the mean map at about 0.55 and moves steadily from there: on the ellipse of 1 : 0.2, t is 0.36 at seven tenths, 0.59 at eight and 0.81 at nine.

Because the share inside a band is flat across stretches of the line, the position reported is the first one along it that comes within a tenth of a point of the line’s best — the least departure from the mean map that does as well. The share itself is not flat in the way that matters: the line’s best is a single number at each band, and the surveyor’s map is one that achieves it.

This is the reading the earlier essay hoped for, in a slightly different form. It does not turn the exponent into a number predicted from the two ends’ distributions; it replaces the exponent with a position on the line between them, found by a search the surveyor can run in the time it takes to count samples. One exponent names one map traded a criterion with many answers for a convex one with one answer at each exponent, and had to ask which exponent; the line keeps one answer for each band and never asks.

Why the ends are the right two #

It is worth saying why this line and not another. Any two conformal maps of a region span a line of conformal maps, and a search along a line through two poor maps would find a poor map. These two ends are the extremes of what a surveyor can want from a band. The mean map is the best available when the band is so tight that only the typical departure matters, and the least worst map is the only one that keeps everything once the band is as wide as w*. Every band between asks for some balance of the two, and a blend of the two is the simplest object that carries a balance.

The balance is not the whole story, as the thin regions show: the best map there departs from both ends in a way no blend of them contains. But Chebyshev’s map is the best at its worst and not on average found the worst-case map and the mean-square map to be different maps on every region but a cap, with every map between them trading range for root-mean-square — and a set of maps running between an average and a worst case is, on the evidence here, very nearly the set of answers a tolerance band can want. The average was a choice of norm put the mean, the mean square and the worst case on one exponent. For a band, the useful axis is shorter and straighter than that: one number between two maps.

The classical criteria — Airy’s mean square of 1861 and Chebyshev’s least worst case of 1856 — were each put forward as the single best map for a region, and which projection is best and every projection minimises something set out why that argument never closed. On this evidence it does not need to: a surveyor with both maps in hand can draw the one between them that the band asks for.

What each number was held to #

A finer sampling must agree. The line’s best map for the square at 0.85 of w* and for the ellipse of 1 : 0.2 at 0.8, re-measured on a sampling four times as fine — 64 rings by 256 azimuths instead of 40 by 160 — must keep the same share to half a point. Both do, so the shares are not an artefact of where the samples fell.

The line contains its ends. At every band on every region its best must keep at least what the mean map and the least worst map keep. It does.

The finding must be there to fail, in both directions. On every compact region the line must come within half a point of the best exponent at every band, and within two points on every region — and the family’s middle members must lie more than a third of their departure off the line on the thinnest ellipse, or the result would be the family restated. The line trails by at most 0.15 points on the compact regions and 1.47 anywhere, and the thinnest ellipse’s members lie up to 0.53 of their departure off it.

Where the line stops #

One line, 201 positions. Between neighbouring positions a finer search could gain a fraction of a point; the comparisons are against twelve exponents, which a continuum of exponents would also improve slightly.

The least worst map is the p = 32 member. The earlier essay found it within half a per cent of the least worst departure on every region, so the line’s far end is very nearly Chebyshev’s map and not exactly it.

The band criterion itself is not the comparison. The earlier measurement found the band’s own search over all maps could beat the exponent family at mid bands, by as much as eleven points on the square, with dozens of answers. Whether the line comes close to that search’s best answers, or only to the family’s, is not settled here.

Area weighted evenly, as before. A readership concentrated somewhere in the region would change which map is best, and the area weighting was a readership all along is the reminder that the even weighting is itself a choice.

Still open: whether a third map closes the thin regions’ gap #

On the thin ellipses the maps the line cannot reach are the intermediate exponents, which leave the line by up to two thirds of their departure. The natural repair is a third end. The members from p = 1.25 to 2 lie farthest off the line on most regions, and the mean-square map among them is the one with a closed-form fit; with it the two ends become a triangle: a plane of conformal maps, every one known once three are solved, searched over two numbers instead of one.

Whether a triangle recovers the last point and a half on the thinnest ellipse, whether the best map inside it for each band lies near an edge or deep in the middle, and whether a surveyor ever needs a fourth, are questions a line cannot answer.