Cartographic Projection
every map is wrong, and by exactly how much
Concept

Degrees of freedom — where it appears

How many independent choices a construction leaves open, which decides whether a condition determines a map, a family of maps, or none. One equation on a two-dimensional map leaves a whole function free, which is why equal-area projections form a family rather than a list.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

Three solutions of one condition, all exactly equal-area. The pseudocylindrical ansatz is x = λ·C(φ) and y = Y(φ) — two unknown functions — and the equal-area condition is one equation, C·Y′ = cos φ. So Y may be chosen freely and C follows, and these three choices give three maps that are equal-area to 1.8e-11 and look nothing like one another: their pole lines are 10%, 100%, 19% of their own equators. That is why the cylindrical family has one equal-area member and this family has as many as anybody cares to name.

The condition does not always decide the map

Write a family as a shape with an unknown function in it and every classical property becomes a differential equation. In three families the equation has one solution and the named projection is what comes back. In the fourth it has a whole function of solutions, which is why that family has forty members and the others have three.

families · Families Where the projection's pole should go, and the answer the third rotation moves it to. Every dot is a pole position the search tried, sized by the best score it can reach there when the third rotation is also free — the score is the distortion of Robinson over Europe by Kavrayskiy's criterion, so smaller is better and the large green dots are the good regions. The circled mark is the two-parameter optimum and the square is the three-parameter one: they are 74 pixels apart on this map, which is a different aspect rather than a refinement of the same one. The search costs 8 times the evaluations of the two-parameter one. Drawn in Mollweide.

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.

choosing · Choosing An equal-area map nobody would publish. Mollweide, with a row-dependent horizontal displacement applied to the page afterwards. That plane map has Jacobian determinant 1 everywhere, so every areal scale factor is untouched: the worst departure from 1 anywhere sampled here is 6.0e-12, which is the arithmetic's own floor. It satisfies the equal-area condition exactly and completely, and it is a ruin. The angular deformation at 30°E 20°N has gone from 11.0° to 60.5°.

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.

choosing · Condition Five stations, ten distances, three spare. A braced quadrilateral with a centre point. Every distance between the corners and every distance to the centre is observed, 10 in all, each with a standard deviation of 8 mm. Holding one station and one bearing leaves 8 unknown coordinates, so the network has three degrees of freedom: three independent statements the observations make that could be contradicted. Everything the adjustment can tell anybody about the quality of the work comes out of those three.

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.

practice · Reduction The part of a request no conformal map can supply — scale falling with distance from the centre. The difference between the requested scale field and the nearest achievable one, over the patch, with the sign shown by colour. The RMS is 1.81e-1 in the logarithm of the scale and the worst single point is 4.91e-1. This is not an error: it is the part of the request that no conformal map of any kind can grant, and its SHAPE is the answer — it says where the request was impossible, which a single number cannot.

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.

choosing · Condition A chain of stations, its positions' uncertainty and its baselines'. Nine stations held at the left-hand one, with every neighbour, second neighbour and third neighbour observed. The filled ellipses are each station's own uncertainty, which grows without limit as the chain runs away from the point it is held at — 7.5 mm at the near end and 574 mm at the far. The open ellipses above each leg are the uncertainty of the baseline, drawn at the same scale: they hardly grow at all, because almost everything that is wrong with one end is wrong with the other in the same direction.

The difference of two coordinates

Three essays give a single coordinate a width. Every practical use of one is a difference of two — a distance, a bearing, a movement, an area — and the width of a difference is not the two widths combined, because the errors are not independent. Far from its datum a one-leg baseline is six times more certain than the positions it joins.

distortion · Precision Every projection against a picture that is not a map. The worst relative distance error over London, New York, Tokyo, Sydney, for each projection in the library and for the best flat arrangement of the same distances. Each azimuthal member is centred on the set's own centroid, which is the fair comparison. The free picture reaches 0.63% and the best projection, Azimuthal equidistant, reaches 8.28% — a ratio of 13.17. The free picture cannot lose, because every projection's own layout was handed to the search as a starting point; what the figure measures is how much the freedom is worth, and it is worth different amounts at different sizes.

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.

impossibility · Embedding The same four requests, put to the two conditions. Each request is a stated field over a square region, and the bar is what is left over after the nearest map satisfying the condition has been found. The conformal condition refuses: its achievable set is decided by boundary values, and the residual is the part of the request no conformal map of any kind can supply. The equal-area condition never refuses — every one of these is met to 3.0e-5, which is the quadrature's own noise — because a positive areal request is granted by a construction with no iteration in it and no boundary data.

The nearest equal-area map to an impossible request

Every number in the previous rung is inside the conformal achievable set, because Liouville's is the conformal condition. The equal-area set is one equation on two functions and never refuses: the same four requests are met to nine parts in a billion, and charged for in angle instead.

choosing · Condition Four layouts, the same five stations, the same ten distances. Every panel has five stations, all ten distances between them, the same instrument precision and the same three degrees of freedom. The ellipses are the error ellipses of the adjusted coordinates, drawn at one common exaggeration, and they are computed from the geometry and the weights alone — no observation value enters any of them. The worst semi-axis runs from 11.1 millimetres to 232, a factor of 20.9, and the difference is entirely where the marks were put.

The network's answer is decided before it is measured

Nine rungs measure what an adjustment does with observations. Every quantity a specification is written about — the error ellipses, the redundancy numbers, the smallest detectable blunder — is a function of the geometry and the weights alone, and does not contain an observed value anywhere. Four layouts of five stations with the same ten distances differ by a factor of 20.9 in their worst coordinate.

practice · Reduction The share of its distances a flat picture can hold. A picture of n places on a plane has 2n coordinates and is unchanged by two translations and a rotation, so 2n − 3 numbers in it are genuinely free; each distance held exactly is one equation. So at most 2n − 3 of the n(n−1)/2 distances can be right, whatever the places are and however hard the map maker tries. The share falls as 4/n: five of six for four places, thirteen of twenty-eight for eight, and 197 of 4,950 — 4.0% — for a hundred. The dashed curve is 4/n, which the count approaches from below and never crosses.

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.

impossibility · Embedding Three named members of the family, and one the family has no name for. The equal-area pseudocylindricals are not a list of maps. They are the solutions of one equation — C(φ)·Y′(φ) = cos φ — in two unknown functions, so one function is free and the named members are points in a space of them. Writing Y as six numbers and searching over them finds a map at 25.03° of mean angular deformation against Eckert IV's 28.08, Mollweide's 31.83 and the sinusoidal's 38.86 — 12.2 per cent better than the best map the family has a name for, and every one of the four is equal-area to the same precision.

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.

families · Families

Named alongside it

The objects these essays reach for when they reach for this one.

ConformalityConstraintDifferential equationEqual-areaLeast-squaresOptimisationAngular deformationClosed formCovarianceEquidistanceError ellipsePseudocylindrical

All concepts