Optimisation — where it appears
Named by 51 essays across 8 fields — each of them below, with the objects they name alongside it.
Every projection minimises something
A projection is the solution to an optimisation problem, and naming the objective explains more than naming the family. Some objectives are exact constraints, some are least-squares fits, and one is a table of numbers a man adjusted until it looked right.
A datum is fitted to a region
Forty countries adopted forty ellipsoids, and the usual explanation is that measurement was poor. It was not — a figure fitted to one country beats the global one over that country by a factor of seven, and what it removes is a systematic bias of two metres in every kilometre.
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.
Fitting the aspect to the region
Choosing a projection is a choice among a few dozen named things. Choosing its aspect is a choice among a continuum, it costs nothing, it changes none of the projection's own properties, and for a long thin country it is worth a factor of 183.
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.
Where the worst point is
The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.
A route that must go round
A shortest route is usually free to go anywhere, and no real route is. The shortest path past a circular exclusion is two tangent great circles and an arc of the rim — a closed form that agrees with a shortest-path search to three metres in 9,598 kilometres — and it costs not the obstacle's size but the square of how far the obstacle reaches past the route.
The average of two projections
The Winkel tripel is literally the arithmetic mean of two other projections, and this site's implementation of it agrees with that mean to the last bit. Averaging beats both ingredients by 26 per cent — and it preserves conformality exactly, destroys equal-area completely, and can turn eighteen per cent of the world inside out without either distortion measure saying so.
The azimuthal family is one function
Five azimuthal projections are taught as five viewpoints — from the centre, from the far pole, from infinity. They are one projection with five choices of how far out to draw a point at angular distance ρ, and each named property is a differential equation in that one function: f′ = f/sin ρ integrates to the stereographic projection and f f′ = sin ρ to Lambert's, from nothing but true scale at the centre.
The rule of thumb, scored
Cylindrical near the equator, conic in the middle latitudes, azimuthal at the poles. It is the most repeated piece of practical advice in cartography and it has never been run against a population of regions. Run against thirty, it is right nineteen times, and every one of its failures has the same shape.
The rule scored out of sample
A replacement rule was read off thirty regions and scored on the same thirty, and this collection recorded that as not being evidence about any other thirty. It is: on forty-five different regions the rule scores 91 per cent against the 83 it managed at home. What it cannot do is the seven regions the collection actually uses, where both it and the rule it replaced name the winner twice out of seven and cost a mean factor of 14.6.
The aspect has three numbers, not one
The site's aspect search has swung the projection's axis through one plane for a long time, and recorded that a region whose long axis runs diagonally has its optimum somewhere that plane never reaches. Searching the whole sphere of pole positions finds 2.6 times more improvement over Japan and 3.4 over the conterminous United States.
The sphere is not the plane at small counts
The site's atlas arithmetic multiplies an ideal sheet count by 2π/√27, the thinnest covering density of the plane. At the four counts whose optimal covering of the sphere is a theorem the plane's number is 21 per cent high at two caps and 9, 5 and 2 per cent low at four, six and twelve — wrong in both directions, and the direction changes with the count.
The third parameter, run
An aspect has three numbers and this site has been searching two of them, with a note admitting it. Searching all three is worth up to 2.1 times — and the obvious way to do it, starting from the two-parameter answer and letting the third move, finds a fraction of that or nothing at all.
The best grid a country could have had
A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.
The shortest route a vehicle can fly
A shortest path under a metric says nothing about whether the thing travelling can follow it. Bound the curvature and the route depends on two headings as well as two positions: the turning cost is a fixed 1.81 kilometres whatever the leg length, so it is 18 per cent of a short leg and 0.28 per cent of a long one, and the whole of it vanishes when the vehicle happens to be pointing the right way.
The landscape the search walks on
The three-parameter aspect search was run and its answer recorded with a note admitting nothing proved it global. Mapping the objective finds 26 to 34 local minima for every projection and region tried, a downhill walk from a random start reaching the best of them 6 to 35 per cent of the time — and one seed from the coarse grid the search already uses reaching it in all four cases. The score is reproducible to two per cent across a sevenfold refinement; the pole it names moves 60 degrees.
Report the map, not the parameters
The aspect search was found returning the same score to 2.3 per cent from poles sixty degrees of latitude apart, which left the answer looking irreproducible. That reading assumed the disagreeing triples make disagreeing maps. They do not — the three answers agree on the distortion field to a quarter of the deformation the map already has.
Sixty zones was a decision about one latitude
Twelve rungs price a grid, a zone, an origin and a reference, and every one of them works inside a single zone. The number of zones has never been asked about: six degrees meets its tolerance at the equator and is loose everywhere else, so a system spending the same tolerance evenly would use 51 zones at the equator and 8 at 82° — and UTM's worst error is 981 parts per million, not the 400 always quoted.
What the net heuristic cannot find
Choosing a net by unfolding outwards from a face beats guessing, and the earlier measurement of that left its own limit unknown. Enumerated in full, the heuristic turns out to be exactly optimal on every solid small enough to check — rank one of 384 — and above that ceiling no sample can tell whether it still is, because eight hundred random nets never reach it.
The shape of the valley
An aspect search returns three numbers, two searches return triples that differ by a hemisphere, and the maps they produce agree. One cause is an exact degeneracy and the rest was called a valley and left unmeasured. Sampled densely, it is neither a valley nor a basin: a connected sheet spanning 170° of pole that splits in two at twice the best score and into thirteen pieces by 1.7 times it.
Where the valley breaks in two
A near-optimal set is one connected sheet at a loose threshold and thirteen basins at a tight one, and the transition was explained without being tested. The explanation is a prediction about region size and projection sharpness: swept over both, the threshold falls from 2.10 to 0.27 as a region grows from 6° to 30°, three projections lie on nearly one curve, and a fourth declines to join for a reason worth having.
The shortest route between two coasts
A shortest path is usually asked for between two points. Between two regions the answer is attained at a pair of boundary points, and which pair is not the pair any page makes look nearest — a cylindrical map picks a pair 2,135 kilometres from the right one across the Atlantic, and 1,716 kilometres wrong between Chile and New Zealand, where its route is 22.5 per cent long.
The nearest map to an impossible request
Rung seven found that not every distortion can be asked for. It never asked what happens when one is asked for anyway — and the answer has a shape: the achievable fields are the solutions of an elliptic equation, a request is a point off that set, and the nearest point leaves a residual whose floor is the curvature rather than the size of the ask.
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 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.
A family is not closed under averaging
A family is a set of maps with a parameter running through it, and the compromise projections here are averages of members. An average of two members is not a member: two conics far apart average to something 31 per cent of their own separation outside the family — and averaging within a family makes the map worse every time, while averaging across two makes it 16 per cent better.
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 basins have widths as well as depths
Measuring the height of the pass left one thing owed: shape means widths too. Measured, the basin has three of them — 32°, 16° and 7° at Japan — it gets wider rather than narrower as the region grows, and the exponent it predicts overshoots the measured one by half again.
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.
The nodes were evenly spaced
Refining an evenly collocated fit improves the solver's report and leaves the map alone. Moving the same number of nodes to the Chebyshev positions — crowded toward the corners, where a conformal map of a polygon is singular — makes the fit settle: 7.030 × 10⁻⁵ at forty-eight nodes and exactly that at every count above it, against an even fit that wanders by 43 per cent and never converges at all.
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.
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.
The map that keeps the most ground inside a tolerance
A surveyor does not 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 and are indifferent to how far outside the rest goes. That is a third criterion, it is answered by a third map, and on a 25° square at half a per cent it keeps 64.1 per cent of the area where the mean-square map keeps 45.4 and Chebyshev's 37.0 — bought by sending a tail out to 5.6 per cent where Chebyshev's worst is 1.5.
A criterion worth using is one whose answer is not unique
Forty-two starts on one region and one tolerance give forty-two different maps, keeping from 36.98 per cent of the area to 67.50. The two starts anybody would actually use — Chebyshev's map and the mean-square map — end 27.1 points apart, which is more than the criterion buys over either of them. And the spread is widest at exactly the tolerance where the criterion is worth using at all, because both facts are the same fact: an objective with a flat top and a sharp edge has somewhere to hide answers.
The pass that fails first is not the one that was measured
Three attempts to explain why the near-optimal aspect set comes apart when it does have fitted the height of the pass between the two deepest basins. That is not the pass that fails first. At every region size the set is held together by the last near-rival to join the best placement's own basin — a basin between 2 and 19 per cent worse than the best — over ground 1.02 to 2.09 times higher than the deep pair's pass. The two passes fall at nearly one rate, −1.42 and −1.31 powers of the region's span, so what separates them is a factor, not a law.
The pass's depth belongs to the projection, and it is not a constant
Separating the fracture threshold into a pass depth and the best score a region admits raised the question of what the pass depth is. Across three projections and four latitudes it runs over a factor of 4.4, and most of that is the projection: Robinson's stays between 0.018 and 0.021 at every latitude, Mollweide's between 0.040 and 0.079. It is not a constant of anything — it falls as the region grows in ten cases of twelve — and it does not follow how much of the projection the region sees. The denominator, over the same cases, moves by half again.
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.
Six projections have two curves, and the refusal has company
Three projections at four latitudes gave three pass-depth exponents, and the question was what sets them. Six at nine latitudes say that most of them have none to set: the Hammer's pass depth rises with what the region sees at R-squared 0.95 and Mollweide's at 0.81, Eckert IV's half-follows, and Robinson's, the Winkel tripel's and the sinusoidal's do not follow it at all. Neither candidate explanation predicts a left-out projection's exponent better than guessing the average — and the sinusoidal, once the lone refusal, is one of three.
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.
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.
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.
The sinusoidal is the edge of its family, not outside it
The sinusoidal is the last member of a family of equal-area maps that begins at Lambert's cylindrical, and it was thought to be the one projection whose pass depth has no curve. Along the family the curve comes and goes: present at m = 0 and 0.8, absent at 0.6 and 0.9. What does move smoothly is the pass depth's level. It is flat from m = 0.2 to 0.9, then rises at every step over the last twentieth, 3.2-fold, to the sinusoid — whose high ridges are the end of that rise, not an exception to the family — while the cylinder at m = 0 stands eight times above its neighbour.
The land puts the homolosine's join eleven degrees higher
Goode joined the sinusoidal to the Mollweide at 40° 44′ because that is where their parallels have the same length — a fact of geometry, not of any objective. On the land his map was drawn for, read from his own lobes, the join has an objective at last, and its best value is 51.7°: a composite joined there gains a fifth of a degree over Goode's, and every join from about 46° to 60° does nearly as well. A smooth curve fitted to the land with no join in it at all stays within three per cent of flat to 50° and then closes, gaining twice as much again. Goode found the right shape and put its knee too low; with Antarctica counted, the land wants no knee at all.
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.
A smooth curve beats the water's best join from its first term
On the land, a smooth equal-area curve with no join needed three terms before it beat Goode's composite rejoined at the land's best latitude, and on the land's own lobes it needs three again. On the water a single term beats the water's best composite, a second and third add more, and the best curve is 0.54 degrees of mean angular deformation better than the composite joined at seventy-five and 1.6 better than Goode's. It is nothing like the composite in shape: its parallels never close to less than seven eighths of the equator's spacing, so the map the water wants is the sinusoidal reshaped by a few per cent at every latitude, not the sinusoidal with a cap. The lobes do not move under it. And where a separate join for each hemisphere was worth three hundredths of a degree, a separate curve for each is worth a quarter of a degree.
The cheapest map that meets its areas
An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
PurposeVerificationObjective functionAspectToleranceRegionScale factorTrade-offAngular deformationClosed formConformalityReproducibility