The average of two projections

Most projections are built. A surface is chosen, a condition is imposed, an equation is solved, and what comes out has a construction anybody can follow.

A few of the most printed maps of the last century were not built that way at all. They were averaged: two existing projections, evaluated at the same point, and the two answers added together and halved.

The Winkel tripel is an average, and the word is not loose #

Oswald Winkel’s projection of 1921 is defined as the arithmetic mean of the equirectangular projection with standard parallel $\varphi_1 = \arccos(2/\pi)$ and the Aitoff projection. The name says so: tripel is German for a triple, and refers to the three distortions the mean was meant to balance rather than to three ingredients.

The site’s own implementation is a transcribed formula that does not look like an average at all — an auxiliary angle, a cardinal sine, two terms over two. Asking whether it is the average is therefore a real question with a possible answer of no.

It is the average, to the last bit a double carries. Sampled every five degrees over the whole sphere, the largest discrepancy between the library’s Winkel tripel and the arithmetic mean of the library’s other two projections is exactly zero.

That is worth having as a check rather than as a remark. The blending machinery was written for this essay and the target was written long before it by somebody transcribing a formula, so agreement means neither is wrong.

What the mixture buys #

The two ingredients score 0.4114 and 0.4290 on Airy’s criterion over the sphere. Every mixture between them scores better than both, and the best — at a weight of 0.45 rather than 0.5 — scores 0.3037.

That is a 26 per cent improvement over the better ingredient, obtained by an operation with no geometry in it whatsoever. It is the strongest available argument for compromise projections as a category, and it is the argument that category is almost never given: the usual defence is that a compromise “balances” the distortions, which is a description of the intent rather than a measurement of the result.

Winkel’s own half-and-half scores 0.3061, which is 0.8 per cent worse than the best mixture on this criterion. That is not a criticism of Winkel. It is a statement about the criterion: Airy’s is one weighting of one pair of quantities, and Winkel was balancing three. A projection built to minimise something else lands near the optimum of Airy’s without being at it, which is the ordinary situation and is what every projection minimises something is about.

Averaging does not always help, and the machinery says which #

The interior minimum above is not a law. Mixing two equal-area projections — Mollweide and the sinusoidal — produces a curve with no interior minimum at all: the best mixture is pure Mollweide, at a weight of zero, and every step towards the sinusoidal makes the criterion worse.

So whether an average is worth taking is a question about the particular pair, and the only way to find out is to sweep it. That is a small piece of engineering with a large consequence for how the family of compromise projections should be read: it is not a technique that reliably improves things, it is a one-parameter search that sometimes has an interior answer.

Over a region rather than the world #

The world is the hardest case and the one compromise projections were invented for, but the operation is not restricted to it, and over a region the numbers change character.

That last sentence is the finding, and it is not an artefact of the criterion. A projection’s distortion has a pattern, and a small admixture of a projection whose pattern is oriented differently can cancel part of it — the same mechanism that makes fitting the aspect to the region worth a factor of 183, arriving through a different parameter.

The gain here is 0.8 per cent, which is small and is not the point. The point is that the sign is positive: the best member of the family is not the better ingredient, even when the ingredients are four times apart in quality.

Conformality survives averaging, and this was a surprise #

The obvious expectation is that averaging destroys exact properties. Mix two projections that preserve angles and the result preserves nothing; mix two that preserve areas and the same.

Half of that is right and half of it is not, and the half that is not is the more interesting.

Every mixture of two conformal projections is conformal. Measured across three pairs and three weights each — Mercator with the stereographic, Mercator with the Lambert conformal conic, the stereographic with the conic — the largest angular deformation anywhere is $2.0\times10^{-6}$ degrees, which is the noise floor of the differentiation and is fifty times below the tolerance measuring instead of naming sets for the property.

The reason is one line once the right words are used. A conformal projection of the sphere is a holomorphic function of the isometric coordinate — that is what conformality is, once the sphere is given its complex structure — and holomorphic functions form a vector space. Any weighted sum of them is holomorphic, hence conformal.

No mixture of two equal-area projections is equal-area. The condition there is that a determinant equals one, which is quadratic, and quadratic conditions are not preserved by averaging at all: the worst areal error across the same kind of sweep is 0.162, which is sixteen per cent and four orders of magnitude above the tolerance.

What that implies about compromise projections as a class #

If mixing conformal maps can only ever produce another conformal map, then no mixture of conformal projections can be a compromise — a conformal map of the whole sphere has unbounded areal error, and no amount of averaging escapes it.

So a compromise projection has to include an ingredient that is not conformal. Winkel’s does: the equirectangular projection preserves nothing and the Aitoff is equal-area in neither of its coordinates. Neither ingredient is exact about anything, which is exactly the freedom the mixture needs.

That is a structural fact about the whole class, obtained from an algebraic property of one of the two conditions, and it is not in any account of compromise projections this site has found.

The failure neither distortion measure reports #

There is a defect averaging can produce that has nothing to do with either exact property, and no quantity this site has defined so far can see it.

The areal scale factor is $|\det J|$ and the two principal scales are square roots. Both throw the sign of the determinant away, so a folded region measures as an ordinary one. The mildest folded point on this map reports 4.98 degrees of angular deformation and an areal factor of 0.517 — numbers that would not raise an eyebrow anywhere on a compromise map.

Only near the fold line, where the determinant passes through zero and the areal factor runs away, does anything conspicuous happen. Away from it the defect is silent, and a projection library that reported the two usual quantities and nothing else would publish this map with a clean bill of health.

Why the fold happens, and when it does not #

A mixture folds when the two ingredients’ derivatives point in sufficiently different directions. Two conformal maps cannot: their derivatives are complex numbers, the mixture’s derivative is the same weighted sum, and a sum of complex numbers is zero only if they oppose exactly — which is a curve in the plane, not a region.

Two azimuthal maps centred 90° apart do, because at points between the two centres one map is pulling east and the other north. Mixing them at any weight produces a region where the combined derivative has passed through zero and out the other side.

That gives a rule with no arithmetic in it: mixing two projections of the same aspect is safe and mixing two different aspects is not, and the site’s own aspect machinery makes the second easy to write by accident. Fitting the aspect to the region shows how much an aspect is worth; this shows what averaging across one costs.

What a mixture is, as a projection #

It is worth being clear that a blend is a projection in the full sense this site uses. It is a map from the sphere to the plane; it has a Jacobian; every quantity here — the two principal scales, the areal factor, the angular deformation, the regional criteria — is defined on it and is computed the same way as for any library member.

What it does not have is a construction. There is no surface it was rolled from, no condition it solves, no viewpoint it represents. That is precisely why the question “is the average of two good maps a good map” cannot be answered by reasoning about the ingredients and has to be measured.

The rest of the class, and what it is not #

Averaging is not the only way to make a projection without a construction. Robinson’s is a table of numbers adjusted until the result looked right, which is a different thing again and is where the indicatrix is a limit finds a projection whose derivative jumps at every table entry.

The distinction worth drawing is about what can be said afterwards. A blend inherits the differentiability of its ingredients, so every quantity this site computes about it is as well behaved as the ingredients’ own. A table with linear interpolation does not, and one of the site’s standing measurements — how large a circle Tissot’s ellipse describes — has no answer at all at a table entry on Robinson.

So the two ways of making a projection without solving a condition are not equivalent. One of them is arithmetic on smooth functions and the other is a lookup, and only the first leaves the machinery intact.

Where the mixture lands in the audit #

Every projection this site holds is measured for angular deformation and areal error and plotted on one chart, and a blend can be measured the same way. The result is where a reader would expect a compromise to be and is worth seeing plotted rather than described: nowhere near either axis.

A compromise projection is therefore not a projection that is nearly conformal and nearly equal-area. It is one that is neither, deliberately, by an amount chosen so that the worse of the two failures is as small as possible. That reading is available from the chart’s geometry and is obscured by the word “compromise”, which suggests a middle rather than a minimax.

What the field says here #

Which projection is best argues that the question is incomplete without a purpose. This essay adds that it is also incomplete without a set to choose from, and the set is larger than the library.

Between any two projections lies a one-parameter family, and its best member is usually neither end. A catalogue of named projections is a discrete sample of a continuous space, and the sampling is historical rather than mathematical: it contains the ones somebody published.

The measurement to make, given a purpose and a region, is therefore not “which of these twenty-one” but “which point in the space these twenty-one span”. That is a larger optimisation and this essay only opens it: two ingredients, one weight, one criterion.

The one thing an average cannot be asked for #

An average has no exact property and therefore cannot answer a question that needs one. That sounds obvious and is routinely ignored, because a compromise projection is pleasant to look at and its failures are nowhere large.

The concrete cases are worth naming. A thematic map whose shading is per unit area needs the areas right, and a projection 30 per cent out in area over the region shades a lie; that is the projection that shows true size and the requirement is exact rather than approximate. A chart on which a bearing is plotted with a protractor needs angles preserved at the point of plotting, and a compromise’s angular deformation of 60 degrees in the corners is not a small residual. And a grid a survey computes in needs a scale factor with a closed form, which an average of two projections has only in the sense that it has a Jacobian.

So the class is exactly what its name says and no more: general-purpose maps, for readers who will not measure anything off them. The measurement in this essay says how good the best such map can be, and the answer — 26 per cent better than either ingredient — is the case for the class rather than against it.

What a mixture costs at the moment it is inverted #

There is one practical price the audit does not report, and it falls on a use every map has: going from the page back to the ground.

A named projection usually has a closed-form inverse, and where it does not — the transverse Mercator’s series, the polyconic — the inverse is a known, published, tested piece of arithmetic. A mixture of two projections has neither. The forward map is a weighted sum of two formulas and is trivial; the inverse of a sum of two functions is not a sum of two inverses, and there is no reason for it to have a closed form at all.

So an averaged projection is inverted numerically, by a Newton iteration on the forward map, one point at a time. That is affordable and it is not free: a few evaluations of both parent projections and their derivatives per point, against one evaluation for a named projection with a formula. Where it bites is the places an inverse is used in bulk — georeferencing a scanned sheet, converting a screen click, warping a raster back, which is a per-pixel inverse.

And the iteration fails exactly where the essay’s fold is. A Newton step divides by the Jacobian, and the fold is the locus where the Jacobian’s determinant passes through zero. Approaching it, the iteration becomes ill-conditioned; at it, the inverse is undefined; beyond it, there are two ground points for one page point and the iteration converges to whichever the starting guess was nearer. That is not a numerical inconvenience — it is the fold reported in the units a user meets it in.

So the fold has a second consequence beyond the one already measured. It makes the map non-injective, which the distortion measures do not report, and it makes the map non-invertible in the concrete sense that the code returns an answer and the answer depends on the initial guess.

The reading is not that mixtures should not be used. It is that the class’s cost is one order lower than the audit shows: the distortion figures are the visible price, and the loss of a formula is the invisible one, paid by every consumer downstream of the map rather than by the person who designed it.

Where this ladder goes #

Two constraints on that larger search are already established. Chebyshev’s criterion gives the exact answer for the conformal maps of a region, so the conformal subspace is solved and mixing within it gains nothing. And a mixture across aspects can fold, so the search space is not simply connected in any useful sense.

What is left is the space between: mixtures of non-conformal projections, over a stated region, under a stated criterion, with the fold as a constraint rather than a surprise. The next rung on this ladder is how many such maps a job needs at all, which turns out to be settled by the tolerance long before it is settled by the projection.