Minimax — where it appears
Named by 17 essays across 3 fields — each of them below, with the objects they name alongside it.
Chebyshev's criterion
The only optimality theorem in the subject. Among all conformal projections of a region, the one with least scale variation is the one whose scale factor is constant on the boundary — a criterion with a proof, a unique answer, and a test anyone can run.
How wrong a flat picture has to be
A determinant proves no flat picture of four places is exact and leaves the size of the failure where nobody can read it. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.
The rule that keeps a route near land is pinned at both ends
A crossing that may never be more than a stated distance from an airfield is refused below one radius and free above another, and each wall is set by a single feature of the ocean. Between them the price falls continuously, with corners and no jumps — and at the lower wall it falls as the square root of the allowance, so the first kilometre is worth twelve of route.
The error belongs to a few of the places
The least error of any flat picture of sixteen cities is 53.3 per cent, and it is not spread over the sixteen. Six pairs hold it there and Sydney is in three of them: leave Sydney out and it falls to 25.5, while leaving out any of five other cities changes it not at all. Choosing which places to draw is worth more than any choice of how to draw them.
A place with a size can be drawn to scale
Points on a sphere have distances no flat picture holds. Places are not points: give every place a radius and the distance between two of them becomes a range, and some flat picture is right about every range once the radius passes a threshold — 38 kilometres for London, New York, Tokyo and Sydney, six metres for five towns in Britain, growing as the cube of the set.
A flat picture has one direction between two places, and the Earth has two
The bearing from New York to Tokyo and the bearing back are not reverses of each other: they miss by 128 degrees. A flat picture draws one line between the two cities, so no picture containing both can have a bearing error below 64 degrees. On any set of places up to about fifty degrees across, the best flat picture sits exactly on that floor — and away from the poles, so does Mercator.
Chebyshev's map is the best at its worst, and not on average
Chebyshev's criterion makes a conformal map's scale constant on the boundary of a region, and no conformal map has a smaller range of scale inside it. The same series, fitted instead to make the scale as nearly constant as possible over the area, is a different map on every region but a cap. On a spherical square Chebyshev's map is 8.3 per cent worse in root-mean-square, the other map's range is 16.5 per cent wider, and every map between them trades one for the other.
Two islands are three numbers, and the proof does not survive them
A sea with one island at its centre has one conformal invariant and a forced best map. Put two islands in it and there are three invariants, readable in closed form because stereographic projection carries circles to circles — and the forcing is gone. The theorem that picked the map needed the region to be unchanged by rotation, not to have one hole, and a single harmonic term two hundredths in size beats the stereographic by 0.19 per cent the moment an island moves off centre.
A second clause decides the half that was already decided
A specification written as a band admits forty-two maps keeping between 37.0 and 67.5 per cent of a region. Adding a tie-break — among those, take the one whose worst departure is least — is a linear programme, and it works: it makes each of the forty-two definite. It also merges three neighbours at the bottom into one, leaves 29.1 of the 30.5 points of spread in area kept, and names a single map only at the tolerance where every map keeps everything and the first clause has stopped asking for anything at all.
One exponent names one map, and at tight bands a better one than the band usually finds
A specification written as a band keeps a count of area and throws the rest of the scale field away, which is why it had forty-two answers and why a tie-break could not name one. Written instead as the mean of |ln k| raised to a stated power, it is convex for every power from one up, and has one answer. At a band of a tenth of a per cent that single answer keeps more of the region than forty-one of the band's forty-two, and at half a per cent fewer than nine of them: the band's best answer leads it by 14.7 points there, and the band's own search started from it reaches within two points of that best. Only as the band approaches the least worst case does a large power take over.
One cut belongs to the island that would cost most alone
A sea with two islands pays a few per cent for being drawn whole by an angle-keeping map, and one cut buys back part of it. Which island to cut is a design choice that leaves the ground exactly where it was, and it is not decided by size: across twenty arrangements of a large island and a small one, the cut worth most goes to the island that would cost the sea most if it were alone in it — in eighteen of twenty, the other two near-ties — which is usually the more central one. A cut joining the two islands buys almost nothing, and the better cut is nearly always the longer.
The second cut goes across the sea
With two islands and one cut, the island that would cost the sea most alone names the cut. With three islands and two cuts, the obvious rule — cut the two dearest — names the best pair in only 23 of 40 seas, and loses up to 23 points of what three cuts would buy. It fails when the two dearest sit on the same side of the sea, where their cuts do each other's work. The best pair keeps the dearest island in 37 of 40; its partner is the island across the centre from it, which is right in 32 of 40 and never costs more than 12 points. A cut joining two islands is never worth a second cut to the sea.
The band names the exponent for a compact region, and not for a thin one
A tolerance band is best served by the mean departure when it is tight and by the least worst case when it is wide. Read as a fraction of the region's own least worst departure, the band decides which, and the region's size drops out: a square, a triangle or a hexagon at 10°, 25° or 40° switches from one to the other at nine tenths, and a surveyor told only that loses at most two points of area on every compact region but the degenerate cap. A thin ellipse does not switch. From six tenths upward it is served best by exponents in between, and any rule with two exponents in it gives up five to seven points there, wherever its switch is put.
Two maps and the line between them do the work of every exponent
A conformal map's log scale is linear in its coefficients, so every blend of the map that minimises the mean departure and the least worst map is itself a conformal map, available the moment the two are solved. Searched along that one line for each band, the blend keeps within 0.17 points of the best of twelve exponents on every compact region and beats all twelve by up to 1.35 on the square; it trails by 1.5 at most on the thinnest ellipse and 0.74 on four regions with a bay, where the rule of switching between the two ends gave up as much as 7.1. The family's middle members lie up to two thirds of their own departure off the line. The line does not reproduce them; it does their work.
Three maps make a plane, and the band's best map lies outside their triangle
Every conformal map whose ln k is a fixed blend of three solved maps — the p = 1 map, the least worst map and the mean-square map — is itself a conformal map, so three solved maps are a plane of them. Inside their triangle the third map recovers most of what the line between the first two gave up on thin regions: 0.63 points short of the best exponent on the thinnest ellipse, where the line was 1.48 short. But the band's best map in the plane is not between the three. On 172 of 182 regions and bands it lies outside the triangle, and it keeps up to twenty points more than any exponent; started from it, a search over all twenty-five coefficients adds about a point on four regions of five and seven on a hexagon. Its price is a worst departure up to nearly five times the least worst.
A sea with five islands is seldom worth five cuts
With three islands in a sea, the second cut went across the sea from the first, and the number of cuts worth spending looked like the number of directions the islands point. With four and five islands that survives in part and breaks in part. Choosing the dearest island and then, each time, the island farthest in bearing from every island already cut names the best pair of cuts in 28 seas of 40, where the two dearest name it in 16 — and it fails at three cuts of five, because the best three cuts often do not contain the best two. The dearest island is no longer reliably among the best pair. And 39 seas of 40 are worth fewer cuts than they have islands: the count follows the widest gap in the islands' bearings more closely than it follows how many islands there are.
A cap on the worst departure keeps the thin regions' gain and gives back the square's
The conformal map that keeps the most of a region inside a band can sit far out in the plane of three solved maps and let its edges go to nearly five times the least worst departure. Capped at twice the least worst, the plane still matches or beats the best exponent on 69 region-and-band pairs of 91: on the thin ellipses it keeps the whole of its gain, because that gain sits in a small pocket just past the exponents' own worst departure and arrives all at once; on the square and the bays it gives most of it back, because there the gain is a ramp that runs out to four times the least worst. Under any one cap the plane still beats the exponents that cap admits, by up to eighteen points — and it can name maps that no weighting of area against worst departure would ever choose.
Named alongside it
The objects these essays reach for when they reach for this one.
VerificationScale factorTolerancePurposeOptimal conformalObjective functionRegionChebyshev's criterionConformalityDegrees of freedomOptimisationEmbedding