Trade-off — where it appears
Named by 44 essays across 9 fields — each of them below, with the objects they name alongside it.
The trade-off is two lines
Conformal means the two principal scales are equal. Equal-area means their product is one. Both at once forces both to one, which is an isometry, which the curvature forbids. That is the entire argument.
A scale bar is right in one place
The bar in the corner of a world map is a picture of a distance, and it is a true picture along one line. On Mercator it reads 500 kilometres for a thousand at 60° north — and on an equal-area map it reads 500 one way and 2,000 the other, so the projection recommended for measuring is the one on which no single correction exists.
The scale factor was chosen
A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.
The square costs the poles
The cut at 85.0511287798° is the most-quoted number in web mapping and is almost never derived. It is where Mercator's northing equals half the world's width — the condition for a square — and it drops 1,901,487 square kilometres. A two-tile root would have reached 89.786° and dropped 3,558.
Giving up continuity
Cutting a map into lobes really does reduce the distortion, by a factor that can be measured. What is paid is that the map stops being one surface — and the size of the tear is the number that pictures of interrupted maps never carry.
Designing a grid for one region
A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.
The pyramid did not have to be Mercator
The usual defence is that a quadtree needs a square world and Mercator supplies one. So does the cylindrical equal-area with standard parallels at ±55.654° — the solution of π cos²φ₀ = 1 — and it needs no polar cut at all. What Mercator actually buys is conformality, and the price of giving it up is 13.8° of shear at 60° north.
Where a pseudocylindrical puts its error
Every projection in this family has to decide what to do at the pole, which on the globe is a point where every meridian meets. Drawing it as a point and drawing it as a line are the two answers, and the trade between them is measurable in both directions.
A grid scaled to the ground is not a map
Construction sites work in coordinates where a tape reading equals a computed distance, which is achieved by multiplying the national grid by a constant. The result is exact at one point, degrades east–west forty times faster than north–south, and is not a map projection at all.
The tolerance decides the model
Every correction inverts to a distance — the line length at which it alone exceeds a stated tolerance. The ranking those distances produce is the same at a millimetre and at a decimetre, and it is different for two jobs at the same tolerance on different ground.
More faces, less distortion, more cutting
The five regular solids are where polyhedral mapping stops, and they stop because there are only five rather than for any reason about maps. Subdivide instead, and the family runs to any number of faces with two fitted exponents: distortion falls as the reciprocal of the count and cutting rises as its square root.
The operation decides the coordinate system
Five candidate planes over one region, scored on the three things a spatial operation depends on. The conformal conic wins shape and distance and is 11.7 per cent out on area; the equal-area member is exact on area and 38.9° out on shape. No candidate is exact in two columns, and no candidate ever will be, because one that was would be an isometry.
A height that is not a length
Level surfaces converge polewards, so the surface a lake sits on is 5.28 metres lower at the pole than at the equator and a height system that reports lengths says a lake runs downhill. The fix reports a number that is constant on the surface and is not a distance from anything: a hundred-metre climb raises it by 99.73 metres at the equator and 100.26 at the pole.
The projections that are beaten on both counts
A rule of thumb has been scored over thirty regions and then forty-five. The same populations answer a harder question nobody has put: which library members are never the right answer at all. Two are beaten outright on both criteria everywhere, one is on no regional front in any population — and it is on the world's.
The net that loses the fewest neighbours
Every one of the cube's 384 nets cuts exactly seven edges of exactly the same length, so the quantity this collection has been pricing cutting by is a constant that cannot choose between them. Measured on the reader's side — how far apart a net puts two places that touch on the globe — the best net scores 11.30 and the worst 17.97, for identical cutting.
Two charts are enough, and one is not
The topological minimum for an atlas of the sphere is two sheets, and the counting fixes a price nobody chose: the worst point of any two-chart atlas is 90° from a chart's centre, where a conformal chart's areal factor is exactly 4 and an equal-area one's angular deformation is 38.94°. A national series has a hundred and twenty thousand sheets, and a hundred and twenty thousand minus two of them are bought by accuracy.
A cell system trades area for shape
A grid can hold every cell to exactly the same area or hold every cell nearly square, and the measurement says it cannot do both: the equal-area cube's areas agree to a part in a thousand and its worst cell is 1.29 times as long as it is wide, while the tangent-warped cube holds shape to 1.19 and lets area vary by 20 per cent.
What a cut buys
Cuts are usually counted and never measured. A cut is a curve on the sphere with a length in kilometres, the shape distortion it removes is a falling function of that length, and the interruption everybody prints spends a hundred thousand kilometres to reach a figure the even-lobed curve reaches with sixty.
A query is a disc, and a disc is not a cell
Everything a cell system does is an address lookup except the one question anybody actually asks it: find everything within five kilometres of here. That is a disc, and the number of cells it fetches is not its area divided by a cell's — at the radii a query is really made at, it is three to seventeen times that.
The address is a curve through the sphere
A database does not fetch a set of cells, it reads ranges of identifiers — so the cost of a query is how many runs its cells form, not how many cells it needs. Hilbert order wins that measurement and loses the one usually quoted for it: its neighbouring cells are further apart in identifier than row-major's, on average and at worst.
An edge has no order of convergence
On a smooth field the three resampling kernels converge at orders 1, 2 and 3 and the choice is obvious. Across a discontinuity they converge at 0.78, 0.58 and 0.60 — within a factor of 1.4 of each other, in an order that puts nearest-neighbour first, and a real raster is mostly edges.
The best flat picture is not a map
A set of dots whose separations are as nearly right as separations can be made beats every projection in this collection — by a factor of thirteen on four world cities, and by eight per cent on sixteen. The collapse between those two numbers is not about cartography. It is that a picture of n places has 2n − 3 free numbers and n(n − 1)/2 distances to spend them on.
The exact map says the seam is smooth
The conformal seam's corner could only be bounded at about two degrees, because the series it was fitted with holds its boundary condition to three parts in a thousand. An exact map exists — the stereographic projection composed with ∫dt/√(1 − t⁴) — and it settles it: the corner falls in exact proportion to the arc the tangent is read over, fitted exponent 0.99, while the gnomonic's 21.4572° is the same to four decimals at every span.
Which projection a weighting can make best
The seven world projections nothing beats on both counts tell a reader with a preference that one of them is their answer. Four of the seven are not: they sit in dents of the front, undominated and unreachable, and no weighting of angle against area can ever put them first. Robinson can be first, on 2.8 per cent of the weight range.
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.
A crossing bends by a law only a conformal chart can show
A ship leaving pack ice for open water takes its quickest route by bending at the ice edge, so that the sines of its angles to the perpendicular stand in the ratio of its two speeds. The law has nothing in it but angles, so on Mercator it can be checked with a protractor — and on a plate carrée at sixty degrees north the same crossing reads as a ship making ten kilometres an hour in the ice rather than six.
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.
The ranking is not an order
A weighting can be made to put almost any projection first. Remove the weights entirely, let each of the seven criteria vote once, and the answer is worse: over the whole sphere Mercator beats the sinusoidal, the sinusoidal beats Eckert IV, and Eckert IV beats Mercator — four such circles, every margin four to three, with a Condorcet winner sitting above them all.
The first break is mostly its denominator
Three earlier measurements have fitted the near-optimal set's fracture threshold against region size and read the answer as a statement about the landscape. It is a ratio, and separating it takes one multiplication: the best score the region admits at all rises with exponent 0.92 and carries 65 to 70 per cent of the threshold's movement on three projections, while the pass's own depth falls slowly and steadily, with exponent −0.42 to −0.49.
A family is a function, not a list
The equal-area pseudocylindricals are the solutions of one equation in two unknown functions, so one function is free and the named members are points in a space of them. Writing that function as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08 — and two numbers are already enough to beat every map the family has a name for.
A ring can be drawn whole, and only one way
A band of latitude has a hole in it, and a region with a hole has a number no angle-keeping map can change. Draw the band without cutting it and that number forces the map: the stereographic from one pole is the least-varying picture there is. One cut along a meridian frees the cone constant, and the scale variation the cut buys back runs from nothing on a polar cap to a factor of 2.13 between the tropics.
The pooled score abandons a region
Every criterion in this subject optimises for one region. An atlas is several, and pooling their samples into one area-weighted score is what everybody does — which on Britain and New Zealand serves Britain 1.2 times worse than it could be served alone and New Zealand 125 times worse. The worst-case objective makes them equal at 33 and 59, and the cost of sharing rises with separation from 1.4 to 59.
The maps with no family are simply better
Seven projections have no continuous symmetry, and they are almost exactly the set anybody would choose for a world map. That coincidence came with a conjecture and a test attached: their advantage should collapse under a criterion that does not care where anything is. Run, it does the opposite — 1.343 times under a uniform weighting and 1.204 under a concentration. The conjecture is refuted.
What the extra unknown costs where nothing can see it
Four lines of position can estimate a common error in every sight as well as the position. The estimate is honest — three kilometres put in comes back as 2.8 — and where the residual is blind, its scatter is plus or minus thirty-one, and the fix that carries it is scattered fourteen kilometres from the ship against the biased fix's two and a half. Letting the residual choose between the two is worse than either, because a test with no power is a one-in-twenty lottery on the worse answer.
One curve beats the three maps it would replace
Goode's composite is a sinusoidal, a Mollweide and now a splice strip between them, and the question left open was whether one smooth spacing curve could do the whole job or whether the joins are doing work no single curve can. One term does it: a curve with a single coefficient carries 6.8 degrees less angular deformation than the composite, two beat every named equal-area pseudocylindrical in the library, and the composite turns out to be worse than the Mollweide it is half made of.
A low sight is worth keeping only if it is weighted
A body five degrees above the horizon is seen through so much air that its refraction correction is ten times a high body's, and a tenth of that correction uncertain doubles the sight's error. Given the same weight as the others, a sight below seven degrees makes the fix worse than throwing it away; weighted by its variance it never does. The ellipse drawn without weights is turned twenty degrees from the right one, and a navigator who must guess the weights loses far less by guessing high.
The seam falls faster than the paper
Divide each face of an icosahedron ν ways and two things shrink at different rates. The fold's error falls as 1/ν, because it is the face's own radius times a degree. The gnomonic seam corner falls as 1/ν², because it needs both a tilt between faces and a distance along their edge. So the gnomonic, 12.8 times the whole assembly budget on twenty faces, goes under it at 720 faces on a 100 mm globe. No globe of any size needs more than 5,120, and what it costs is paper folded out of sight: 5.8 per cent of the sheet at that crossing.
Scored lobe by lobe, the homolosine loses to its own lower half
Read as one uncut sheet, a single smooth curve beats Goode's composite by 9.6 degrees. But the homolosine is used cut into six lobes, so here it is scored cut, with each point measured from its own lobe's meridian, and the ranking turns over. Eckert IV, the best named map uncut, falls to second from last. The curve that beat the composite is now 2.3° worse than it. The plain sinusoidal, the composite's own lower half, wins at 17.88° against 18.48. The best smooth curve for a cut map is the sinusoidal with a few per cent of correction.
The ocean centres its lobes on the land's cuts, and cannot keep its own cuts short
Run on the water instead of the land, the same exact search gives an interrupted homolosine whose lobes sit on the Pacific, the Atlantic and the Indian Ocean — five of its six central meridians within twenty degrees of a cut the land's own layout chose — and it deforms the water by 13.31° against the 19.75° the land's layout gives it. The inversion stops at the cuts. Every meridian on Earth crosses at least 7,172 kilometres of water, so the ocean's map has no layout whose cuts stay under seven thousand, where the land's went below nine hundred. And no layout serves both: none comes within 2.57° of both optima at once, though a single wide Pacific lobe buys the water 2.8° for a tenth of a degree of land.
The water would join Goode's map at seventy-five degrees
Goode joined the sinusoidal to the Mollweide at 40° 44′, and the land, given its own lobes, moves that join up to sixty degrees. The water moves it further, to seventy-five, and gains twice as much by it — a full degree of mean angular deformation against the land's half — though less of the northern water than of the northern land lies above Goode's join. The gain is in the polar water, north and south, where Goode's Mollweide cap shears even the lobes' own meridians; the water just above his join, the northern edge of the Southern Ocean among it, pays part of it back. A separate join for each hemisphere is worth a few hundredths of a degree, and with the join free the one-layout-for-both question comes out the same way it did with the join fixed, only worse: no layout comes within 3.3 degrees of both surfaces' best at once.
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.
The best compromise for angle is not the best for bending
Every compromise projection in the library is an average of two others, and averaging is a first-order operation — so the second derivative was never part of the bargain. Swept along five ordinary blend paths, the weight that minimises angular deformation and the weight that minimises flexion are between a quarter and a half of the axis apart, and how much a compromise buys at one order predicts nothing about the other.
One number changed and the whole map moved
Raise one bump's weight in a density specification, leave every other number identical, and solve again. London's own value is what changed; London moves 0.054 and Delhi moves 0.226 — four times as far, with its own number untouched. The largest displacement anywhere is thirty degrees from the change, and the antipodal band still moves a fifth of the peak.
The lines where the bending vanishes
Every projection has a line printed in its margin — the standard parallel, where the scale is exactly one. It has a second special line nobody prints: the one along which a geodesic is drawn straight to second order. On Gall–Peters they are forty-five degrees apart, and the rule turns out to be tangency — a tangent construction puts both lines at its point of contact and a secant one moves only the first.
Named alongside it
The objects these essays reach for when they reach for this one.
VerificationPurposeToleranceAngular deformationEqual-areaOptimisationClosed formScale factorConformalityPareto frontAggregationConvention