Constraint — where it appears
Named by 25 essays across 6 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 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.
A projection written as a condition
Instead of a formula, a sentence: the distance from these two places must be exactly right. The map that satisfies it is found by intersecting two circles, it is exact to five parts in a hundred million million, and it exists over the whole sphere for a reason that belongs to the sphere rather than to the construction.
A map that cannot be read backwards
Craig's projection answers one question exactly — lay a straight edge from any place to the centre and read the compass course, right to 6 × 10⁻¹⁴ of a degree. It pays by folding: 78°S and 48°S on the same meridian are drawn at the same point, so no inverse exists and nothing else can be read off it at all.
Solving for the map instead of choosing it
Chebyshev's criterion has sat on this site since its second phase with one case it could be applied to: the spherical cap, whose answer is the stereographic projection. For any other region the site stated the criterion and stopped. It is a linear least-squares fit, and the fitted map beats every named projection over the region it was fitted to.
Every equal-area map is every other one
Take Mollweide and slide every row of the page sideways by an amount that depends on the row. The result satisfies the equal-area condition to 6 × 10⁻¹², exactly as well as Mollweide does, and it is a ruin — the angular deformation at one ordinary point has gone from 11° to 60°. Equal-area is one equation, and one equation leaves a whole function free.
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.
Not every distortion can be asked for
Six essays have written projections as conditions and asked how much freedom a condition leaves. The reverse question has never been put: a cartographer knows what distortion they want, so can they ask for it? For a conformal map the answer is a single equation, it is the Theorema Egregium in disguise, and asking for no distortion anywhere fails it by exactly the curvature of the sphere.
A coordinate is the output of a solve
Six essays measure a tape, close a traverse, spread a misclosure and reduce a chain. The coordinate that comes out of the far end is the solution of a least-squares problem, and the problem has a decision in it that is not a measurement: what to hold fixed. Change it and every coordinate moves by centimetres while not one residual moves at all.
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.
Five distances of six, and never more
A flat picture of n places has 2n − 3 free numbers and n(n−1)/2 distances, so it can be exactly right about 2n − 3 of them and no more — five of six for four places, twenty-nine of a hundred and twenty for sixteen, and 3.98 per cent for a hundred. The bound is reached by a construction with no fitting in it, and an azimuthal equidistant map spends only n − 1.
The escape is not a dimension
Four places that will not lie in a plane surely lie in a space — and they do not. The double-centred matrix of their great-circle distances has a negative eigenvalue, so no Euclidean space of any dimension holds them, and buying a third dimension improves the picture by nothing whatever, to fifteen decimal places. What does work is a page with curvature, and the curvature is pinned to within two per cent.
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 map depends on where it was cut
Solving the discrete conformal equations on a triangulated body gave an areal spread of 3.07, with the number possibly belonging to the patch, the boundary and the two pinned vertices rather than to the surface. It belongs to the pins: hold a different pair and it runs from 2.68 to 7.44, while the typical angular deformation does not move at all.
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.
Five bearings of twelve, and only eight ways to draw them
A flat picture of four places can hold five of their twelve one-way bearings exactly, the same count as distances, and the five are found by solving equations rather than searching. The equations cannot tell a line from its reverse. Of the 192 ways to choose five, eight can be drawn with every arrow pointing forwards, all eight leave London and New York unheld, and on sixteen cities no choice made at random can be drawn at all.
The pair that cannot be held is not the one the geometry names
Four world cities have exactly one pair of the six that no drawable picture holds, and the obvious candidate — the pair whose bearing there and bearing back disagree most — is not it. That quantity predicts something else exactly: whether a held pair's arrow has a forced direction. It separates the two groups with a clean gap and nothing in between. And at five places there is no forbidden pair at all, so the four-place case was a statement about how little room five constraints leave rather than about the map.
A mixed picture holds one more, and what it is holding is north
A flat picture of n places can hold 2n − 3 distances exactly, and it can hold 2n − 3 bearings exactly, and it is tempting to read one budget spent twice. It is not one budget. A distance cannot see which way the page is turned and a bearing cannot see how large it is drawn, so a picture constrained by both kinds is blind to one freedom instead of two — and holds 2n − 2. For four cities that is six of the eighteen available where either kind alone can hold five, and nothing whatever can hold seven: nought of 31,824 choices.
An equal-area strip removes the corner and charges nothing for it
Goode's seam matches three quantities and breaks the fourth: every meridian but the central one turns a corner there. A third map spliced between the two removes it — four conditions on one function, satisfied uniquely by a cubic — and the expected price does not arrive. The spliced composite carries less angular deformation across the seam than the plain join, not more: 34.7° against 37.7° at sixty degrees of longitude. What it does leave is the same defect one derivative up, a jump in curvature falling as the reciprocal of the strip's width.
Two numbers about the paper buy nothing about the places
A picture of n places has four freedoms, and a constraint holds only what its own blindness leaves: distances reach 2n − 3, bearings 2n − 3, the two together 2n − 2. Position is reached by neither, because nothing about how places stand to one another says where on the sheet to put them. A stated position does reach it, and two of them take the budget to 2n exactly. What the count conceals is that the extra two buy nothing: a largest picture holds six place constraints with the pins and six without, the same 7,596 of them, holding the same pairs at the same rates. The constraint system is a direct sum, and a pin is a different currency.
A straight baseline moves more water inward than sea outward
The seventh article of the law of the sea lets a coast fringed with islands measure its territorial sea from straight lines joining chosen points, and gives no length, no angle and no scale for the choice. On a stated coast the lines a common reading permits add sixty-eight square kilometres of sea at the outer edge and turn nine hundred and eighty-eight into internal waters. Drop the test of direction and the outer rocks come into reach: eighteen hundred added, forty-six hundred enclosed.
Only Eckert II's family can be the strip, and its meridians are straight
The cubic that smooths Goode's seam is four conditions on the length of a parallel, and equal area turns them into four conditions on how the parallels are spaced. The sinusoidal's spacing is flat, so a strip can leave it only where its own spacing is momentarily flat — and of eleven named equal-area maps, only Eckert II's spacing ever turns over. One member of its family meets all four conditions exactly, over seventeen degrees centred four hundredths of a degree from Goode's join, with every meridian straight. The cubic over the same span is the same map to four parts in a hundred thousand.
A second pin is a measurement of the places
Counted exactly, a picture of places is a direct sum: the distances and bearings spend its shape and two pins spend where it sits on the sheet. Given tolerances, the sum survives in one form and fails in another. Pins at one place put none of their error into the shape at any tolerance, and wherever the pins are, their average takes none of the distances' error. But every place pinned after the first is a statement about the geography, and held exact it makes the shape worse once its error passes about two thirds of the shape's own.
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.
Written in numbers, the archipelago rule still leaves eighteen
The article on archipelagic baselines gives its tolerances as numbers: no segment over a hundred nautical miles except three per cent of them, which may reach 125, and water between one and nine times the land. On a stated archipelago 8,799 closed systems keep the main islands inside, the ratio passes all of them and the length test eighteen, and the eighteen still differ by 27,585 square kilometres. Thin the land and the ratio does the deciding; the lower bound decides nothing; and the three-per-cent exception can be used only by a system padded to thirty-four segments.
Named alongside it
The objects these essays reach for when they reach for this one.
VerificationDegrees of freedomClosed formConformalityEmbeddingGreat circleOptimisationConditionLeast-squaresPurposeAzimuthBearing