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

Pseudocylindrical — where it appears

A projection with straight, evenly divided parallels and curved meridians, whose equal-area condition leaves a whole function free. The free function is the pole line's length, and choosing it is what separates Mollweide from Eckert IV from the sinusoidal.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

What a pole line buys, and what it costs. four pseudocylindrical projections placed by the two numbers the decision moves. Across the bottom, the angular deformation averaged over the whole sphere, where the pole-line projections — Eckert IV and Robinson — are the better maps. Up the side, on a log axis, the factor by which the last parallel is stretched, where they are worse by more than tenfold: 49× at best against 4.6× at worst for a pole drawn as a point. A pole line represents one point of the globe by a line of the map, and that is the price.

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.

families · Families 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 What a cut buys. The mean angular deformation of the interrupted sinusoidal against the total length of cut the interruption spends, for lobe counts from one to twenty-four. Goode's interruption — the one actually printed — is the marked point: it spends 100 thousand kilometres and returns 18.0°, where the even-lobed curve returns 9.2° for the same length. It is not on the frontier and it was never trying to be: its cuts are placed to keep continents whole.

What a cut buys

Six rungs count cuts and none measures one. 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.

impossibility · Topology What is left after the best rigid motion of the page. Every projection in the library, rotated about its own symmetry axis by five degrees, with the best rigid motion of the page fitted and the leftover measured against the picture's own size — on a logarithmic scale, because the answers span four orders of magnitude. Fourteen sit at 2e-6 or below, which is the axis search's own floor. Seven sit between 9e-3 and 3e-2. There is nothing in between, so the split is a fact rather than a threshold — and the fourteen are exactly the cylinders, the cones and the planes.

The family is a symmetry, not a shape

Rung seven dissolves the taxonomy: eleven named maps fitted to one perspective formula, six reproduced exactly and five refused, so the developable surface describes how the projections were discovered rather than what they are. What is left is the question it does not ask. If the shapes are not the classification, what is?

families · Families Three implementations of one projection, and three different distortions. The spread in angular deformation between linear, Catmull–Rom and Aitken interpolations of Robinson's own table, at 60° of longitude. It is exactly zero at every fifth degree, because all three pass through the tabulated rows, and it is not zero anywhere else: 37 of 87 latitudes differ by more than half a degree and the worst is 8.42°. The graticules the three draw differ by 4.2 parts per thousand of the map's own span, which is under the width of a printed line.

A projection defined by a table has an interpolation in it

One member of this library has no formula: Robinson set nineteen pairs of numbers by eye and the table is the definition. Three published interpolations of it draw graticules that differ by four parts in a thousand of the map's span and report angular deformations that differ by 8.4 degrees.

families · Families 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.

Angular deformationEqual-areaDegrees of freedomPole lineTrade-offAspectClosed formCone constantDifferential equationGoode homolosineOptimisationProjection family

All concepts