A family is a function, not a list

Ten rungs of this anchor treat a family as a set of maps with a parameter running through it. The conic family has a cone constant; the azimuthal family has a radial function chosen from five named ones; the cylindrical family has a standard parallel. In each case the family is presented as a list with a dial on it, and the dial is one number.

The condition does not always decide the map is the rung that notices this is not true of every family. Write a family as a shape with an unknown function in it and every classical property becomes a differential equation; in three families that equation has one solution and the named projection 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.

The fourth family is the pseudocylindricals, and this rung takes the observation seriously.

The family is one equation in two unknowns #

A pseudocylindrical projection is x = λ·C(φ), y = Y(φ): parallels are straight and horizontal, meridians curve, and the whole construction is two functions of latitude. C says how long each parallel is drawn and Y says where it goes up the page.

Imposing equal area gives one equation:

C(φ)·Y′(φ) = cos φ

That is the whole family. One equation in two unknowns leaves one function free, so choosing Y decides C and any Y at all gives an equal-area map — which is the point the condition does not always decide the map makes and the reason the machinery for it already existed in this collection.

The named members are choices of Y. The sinusoidal takes Y = φ, which gives C = cos φ and parallels true to scale. Mollweide takes the Y that makes the outline an ellipse. Eckert IV takes one that gives the pole a line rather than a point. Each is one function out of infinitely many, and nothing about the equation prefers any of them.

Putting coordinates on the family #

A function is not a list of numbers and cannot be searched. Giving the family coordinates means writing Y in a basis, and the choice of basis is where two attempts failed before one worked — both for the same reason, and the reason is the pole.

The basis that works is

Y(φ) = a₀·(2φ/π) + Σ aₖ·sin(2kφ), k = 1 … K

Every term is odd about the equator and zero there, which a pseudocylindrical requires. The linear term sets how far up the page the pole goes. The sine terms shape the spacing of the parallels between the equator and the pole and vanish at both ends, so they move the pole’s height not at all.

The obvious basis is the odd harmonics and it cannot hold Eckert IV. A series in sin((2k−1)φ) is orthogonal on this interval, which makes fitting trivial, and every one of its terms has derivative zero at φ = π/2. So it forces Y′(π/2) = 0, which makes C = cos φ / Y′ the indeterminate 0/0 at the pole and refuses every map with a pole line. Eckert IV has one. A first version of this machinery could rebuild no named member at all.

Using all the harmonics repairs that and loses the orthogonality, because sin(jφ) and sin(kφ) are not orthogonal on a half-period interval. Projecting onto a non-orthogonal basis by an integral gives coefficients that are not the fit, and the reconstructions came back with Y decreasing over most of the sphere — which is not a map, it is a page folded over itself.

So the coefficients here are a weighted least-squares fit rather than a projection, and the weight is cos φ, so the equatorial half of the map decides the fit rather than the last few degrees before the pole.

The check that the coordinates are coordinates #

Two things say the parameterisation is on the family rather than on something adjacent, and both are cheap.

One point is known exactly. Set a₀ = π/2 and every other coefficient to zero: Y = φ and C = cos φ, which is the sinusoidal. Scored through the machinery it gives 38.860646°, and the library’s own sinusoidal gives 38.860646°.

The other named members are reproduced to a stated residual. At six terms Mollweide’s shape function is held to 6.85 parts in a thousand of the map’s own height and Eckert IV’s to 6.93, which is close and is not exact. So the comparison scores the named maps as themselves rather than through the basis, and the residual is reported beside them. Nothing in the result depends on how well the basis holds a named member; the basis is only used for the searching.

Two free numbers already beat every name #

Once the family has coordinates, searching it is a pattern search over a handful of numbers, and the interesting result is how few are needed.

Between one and two. That is the result worth carrying: the best two-parameter member of this family is better than every member it has a name for, and the second parameter is worth 11.3 degrees while the third is worth 1.3 and the fourth 0.3.

The one-parameter row is a small surprise of its own. Y = a₀·2φ/π gives C = (π/2a₀)·cos φ, which is the sinusoidal with its two axes scaled by reciprocal factors — still equal-area, because the product of the scalings is one, and not the same angles, because an anisotropic scaling is not a similarity. The best a₀ is 1.635 rather than π/2 = 1.571, and it scores 38.36° against the sinusoidal’s own 38.86.

So even before any shaping, the sinusoidal is drawn at the wrong aspect ratio for the criterion it is scored on, by a little over one per cent.

The search, and what stops it wandering out of the family #

A coefficient list is not automatically a map, and two conditions separate the ones that are.

Y must be increasing. If Y′ goes negative anywhere, the page folds over itself — two latitudes drawn at the same height, with the ground between them inside out. The search checks Y′ at a hundred and eighty-one latitudes on every candidate and rejects any that dips below a thousandth.

C must be finite. C = cos φ / Y′, so a zero in Y′ sends a parallel’s drawn length to infinity. That is the same condition as the first with a sharper edge, and it is why the guard is a small positive number rather than zero.

Those two turn the search into a constrained one, and a pattern search handles a constraint the crude way — by scoring an infeasible candidate as infinite and never stepping there. That is the right crudeness here, because the constraint is not a boundary the optimum sits on: the three-number optimum has a minimum slope of 0.348 and the six-number one 0.174, both well clear of the edge and neither pressed against it.

What the searched map is not #

The map at the top of this essay is better than Eckert IV on one number and that is the whole of what has been shown. Four things it is not.

It is not better on every criterion. The score here is area-weighted mean angular deformation over the whole world, which is one of the seven criteria this collection computes and is the one Eckert IV was built around. Which projection is best establishes that the ordering of projections is a function of the criterion and the region, and nothing here escapes that: a search that minimises a different number finds a different map.

It is not a better map to look at. Nothing has been optimised about the shape of the outline, the behaviour at the corners, or whether the graticule reads pleasantly — and a projection defined by a table has an interpolation in it is the rung about Robinson, which is the family’s reminder that a map chosen by eye can be a perfectly good map and impossible to write down.

It is not new. A search over a function space finds what the space contains, and this space has been searched before by people who then named what they found; the named members are what previous searches over previous criteria produced. What is new here is doing it with the family stated as a function space rather than as a list.

And it is not the optimum. The six-number search reaches 25.03° and the ladder is still falling; the family is infinite-dimensional and this is a pattern search over six coordinates from one starting point. What the number is, is a bound from below on the family — the family contains at least one map this good — which is exactly the role the best flat picture is not a map gives a free optimum in another field.

What this says about the other families #

Nine rungs have taken the families apart as surfaces, as one-parameter constructions, as symmetries and as conditions. This one adds a dimension count, and the count separates them in a way the earlier rungs did not.

The conic family has one free number, the cone constant, and the conic is the whole family shows the cylindrical and azimuthal cases are its two exact limits. Searching it is searching a line.

The azimuthal family has one free function — the radial function f(ρ) — and the azimuthal family is one function establishes that each named property is a differential equation in it with a unique solution. So the family is infinite-dimensional and every condition anybody imposes on it picks out a single point, which is why it has five named members and no search has ever been run over it.

The pseudocylindrical family has one free function and its defining condition uses up the other, so the condition picks out a sub-family rather than a point. That is the structural difference, and it is why this is the family with forty members: it is the only one where imposing the property everybody wants still leaves a function to choose.

Which suggests a question the anchor has not asked. If the azimuthal family’s conditions each pin it down completely, is there an azimuthal condition weak enough to leave a function free — and if there is, does that family have forty members nobody has named either?

The criterion, and what happens on a different one #

Every number above minimises area-weighted mean angular deformation over the whole world. That is one of seven criteria this collection computes, and the honest question is how much of the result is about the criterion.

Some of it, and the amount is measurable. Run the same search against scale spread — the ratio of the largest principal scale factor to the smallest, over the world — and the named ordering inverts completely: the sinusoidal reads 11.11, Eckert IV 14.81 and Mollweide 16.68, so the map that is best on angular deformation is the middle one here and the map that is worst there is the best here. The searched member reaches 5.92, which is 47 per cent below the best named map rather than twelve.

So the criterion decides both which named map wins and how much room the family has above it. What does not depend on the criterion at all is the structure of the result — that the family is a function space, that its named members are points in it, and that a handful of free numbers leaves every name behind.

What does depend on it is the twelve per cent. That number is the distance between the best named map and the best searched map on one axis, and which projection is best is the essay establishing that such distances are not transferable: the same two maps compared on the mean scale departure and on the worst case have rank correlation −0.04 across the library.

So the claim this rung makes is narrow and is not weak. On the criterion Eckert IV was designed around, the family contains a map twelve per cent better than Eckert IV, and it contains it in a part of the family that has no name — which says something about naming rather than about Eckert IV.

What is left free after the search #

The search here optimises a shape function against a criterion and holds everything else fixed. Two things it holds fixed are themselves free.

The aspect. Every map here is drawn in the normal aspect, poles at the top and bottom, and the aspect is a free choice prices what rotating a projection’s axis buys and finds it the cheapest available improvement. A pseudocylindrical in an oblique aspect is a different map with the same shape function, and the two freedoms are independent.

The interruption. Giving up continuity is the rung about cutting a map into lobes, and Goode’s homolosine is the pseudocylindrical family’s own answer to it — two named members joined at a latitude and interrupted. That is a third freedom: which member where, and where to cut.

Three independent freedoms, only one of which this rung searches, and the searched one turns out to be worth twelve per cent. What the other two are worth in the same units is not known here.

What a family with a dimension is for #

Reading the rung back, the useful thing it produces is not a map. It is a number attached to a family: how many free numbers its members have.

That number decides what can be done with a family. A one-dimensional family is a dial and the right thing to do with a dial is sweep it, which is what the conic is the whole family does and what every parameter figure on this site does. An infinite-dimensional family with a condition that pins it is a menu, and the right thing is to read the conditions off. An infinite-dimensional family with a condition that does not pin it is a search space, and the right thing is to search it — which nobody had done here.

The three cases are told apart by counting equations against unknown functions, which takes a line of algebra and no computation:

Equal area is one equation. So the cylindrical, azimuthal and conic families each have their free function determined by it and the pseudocylindrical family does not, and that single count is why one family in the taxonomy has forty named members and the others have three.

It also says which other families are worth searching, and the answer is the ones with two unknown functions. This library holds one other: the polyconic shape, x = λ·C(φ, λ), whose parallels are not straight — which is more freedom again and is the reason the projections that gave up being one thing finds a map preserving nothing the usual tests look for.

The seven that have no family at all #

Ten rungs have asked what a family is: a construction, a condition, a symmetry, a definition by table, and now a function space with a dimension.

Every one of those accounts has the same gap and the family is a symmetry, not a shape names it. Seven projections in this library have no continuous symmetry at all, so they are not members of a family in any of these senses — and they are Robinson, Winkel tripel, Eckert IV, Mollweide, sinusoidal, Hammer and the polyconic, which is very nearly the set of projections anybody would choose for a world map.

That is stranger after this rung than before it. Two of those seven have just been shown to be points in a perfectly ordinary function space, so they do belong to a family — a family with no symmetry in it. The projections with no family are the ones people use, and the anchor still has no account of why.