The best compromise for angle is not the best for bending

The average of two projections establishes that a compromise is worth making: blending two maps buys more than blending their errors, so the angular deformation of a half-and-half blend sits well below the average of the parents’. That is why compromise projections exist, and it is why the Winkel tripel — which is literally the average of two other library members — is on more atlas endpapers than any projection of the last century.

Every word of that argument is about the first derivative. Averaging two maps averages their Jacobians exactly, and every first-order quantity follows.

Nothing in it says anything about the second derivative, which is what this anchor measures.

Why the second order need not follow the first #

The Jacobian of a blend is the blend of the Jacobians — that is one line of differentiation, and it makes every first-order measure an interpolation of two known values, so a compromise cannot be worse than both its parents at first order.

Flexion is not an interpolation. It is a second difference divided by a first: the rate at which a drawn geodesic turns, per unit of the length it is drawn at. The numerator interpolates and so does the denominator, and a quotient of two interpolations is not an interpolation.

Two consequences follow, and only one of them turns out to be large.

The small one is that a blend can bend more than both its parents, wherever the denominator dips faster than the numerator — which happens where the two parents’ Jacobians partly cancel.

The large one is that even where the blend beats both, how much it beats them by is unrelated to what it bought at first order.

The first-order picture, for comparison #

The curve above is the reason compromises exist and it is complete as far as it goes. What this rung adds is a second curve, on the same axis, which nobody draws.

What a compromise buys, at each order #

The first row is the extreme case and it is worth reading slowly. Blending Mercator with the Lambert cylindrical equal-area buys a 35 per cent reduction in angular deformation against what the average of the two errors would be — a very good compromise by the standard the ladder below this one uses — and buys 4.5 per cent in bending.

The last row goes the other way: blending Mercator with the sinusoidal buys slightly more at second order than at first.

A spread of nine to one across five ordinary pairs is not a correction. It means the first-order gain, which is the number everybody computes, carries no information about the second.

It also inverts the intuition about which blends are good ones. Mercator with the Lambert cylindrical is the pairing that looks most promising from the first-order column — a conformal map and an equal-area one, each carrying the whole of a sec²φ in a different place, so the average cancels most of both — and it is the pairing whose second-order gain is smallest. The cancellation that makes the first order so good is a cancellation of two scales, and the two curvatures do not cancel with it.

The two optima are half the axis apart #

The sharper form of the same finding is that the weight differs.

The angular optimum sits between 0.13 and 0.38 on these five pairs; the flexion optimum sits between 0.50 and 0.75. They never coincide and they are never close.

A published compromise names one weight and does not say which of the two it was chosen for. Nor could it have: the second-order measures are forty years younger than most of the compromises in any atlas, so the weight that was chosen was chosen against the only criterion available. Winkel’s is a half-and-half average of the equirectangular and the Aitoff, chosen in 1921 by a judgement about how the result looked; the first-order machinery that would justify it did not exist and the second-order machinery still does not get used.

The rare case the argument predicted #

The quotient argument does produce a blend that bends more than both its parents. It produces exactly one, in a hundred and twenty pairs.

One point three per cent, once in a hundred and twenty. The pairs swept are every pair drawn from sixteen library members that can cover a thirty-degree cap, at nine weights each, which is a hundred and twenty paths and a thousand and eighty blends. The prediction was correct and the effect is negligible, and saying both is more useful than saying either — a mechanism that is real and small is a mechanism to stop looking for.

That is a result this rung was written to find and did not. The plan for it expected overshooting to be the headline, on exactly the reasoning above, and what the sweep found instead was the decoupling. The reasoning was sound and the magnitude was a guess.

The parents, measured at both orders #

That earlier rung’s finding is the precondition for this one. If the two orders ranked the library alike, a blend’s second-order behaviour would be predictable from its first-order behaviour and there would be nothing here. They do not, and the disagreement between the two rankings is what the blend path inherits.

Where the flexion actually goes #

There is a second reading of that field worth having. A blend’s flexion is not the blend of its parents’ flexions, so the field above cannot be assembled from the two parents’ fields by any weighting — it has to be computed. That is a statement about cost as much as about mathematics: a designer sweeping a blend path pays for a full second-order sweep at every weight, where the first-order sweep is two evaluations and an interpolation. The reason the second curve is missing from the literature may be nothing more interesting than that it is a hundred times dearer to draw.

That is the geometric reason the two optima differ. The first-order error of a blend is smallest where the two parents’ errors most nearly cancel, which is where they point in opposite directions. The second-order error is smallest where the parents’ curvatures cancel, and there is no reason for those two places to be the same place — curvature is a different function of position from scale, and on these library members it is a differently shaped one.

What the check refuses #

Two controls, and both are about the ends of the path rather than its middle.

A blend at weight zero is a parent. Its ratio to the straight line between the parents is exactly one, by construction, and a measurement that did not return one there would be measuring its own interpolation rather than the map. It returns one to the last digit at both ends of every path.

And every pair has to beat the line at first order, or it is not a compromise and the comparison has no subject. All five do, by between 19 and 35 per cent — which also says the parents were chosen as ordinary pairs rather than as pairs that make the finding come out.

The one thing the sweep cannot refuse is its own resolution. Nine weights place each optimum to within a step of 0.125, and the two optima are between two and four steps apart, so the separation is resolved and its exact value is not. Doubling the sweep moves each optimum by under a step and leaves the separations where they are.

What a designer should do with this #

Three things, and the first is nearly free.

Report both. A compromise projection is presented with one figure of merit, and computing the second costs one more sweep of the same machinery — the same second-order field the anchor has been computing since its first rung. A designer who knows that the flexion optimum is at 0.5 and the angle optimum at 0.13 can choose between them; one who has only the second number cannot know there was a choice.

Choose by purpose, which is this field’s standing rule. No essay here may say a projection is best without naming the purpose, and the two orders answer different purposes: angular deformation is what a reader estimating a shape is affected by, and flexion is what a reader laying a straight edge on the map is affected by — which is the ruled-line limit and is a real operation on a real chart.

And do not blend to a rule of thumb. Half and half is the weight every published compromise uses, and it is not the optimum for either order on any of the five pairs here — it is between the two on three of them and outside both on two.

What a reader is actually doing with the map #

The two orders are not two spellings of one quality, and the difference is visible in what a reader does.

Angular deformation is what a reader estimating a shape is affected by. A country’s outline, a lake’s roundness, whether two coastlines are parallel — all of those are read at a point, from the local map, and the local map is a linear one. A projection with small angular deformation gives a reader shapes they can trust in the small.

Flexion is what a reader laying a straight edge is affected by. A ruled line across the sheet is a claim about a route, and the amount the true route bows away from it is the flexion times the length squared over eight, which is the ruled-line limit this anchor measures. A navigator, a planner drawing a corridor, anybody reading a distance off a scale bar along a line — all of them are using the second derivative and none of them is using the first.

So the choice between the two optima is not a matter of taste between equivalent summaries. It is a question about which operation the map is for, which is this field’s standing rule and which nobody has ever been in a position to apply here, because only one of the two numbers has ever been computed.

The awkward case is the atlas endpaper, which is used for both and is chosen once.

It is also worth noting what the two optima share. Both are interior, both are single-basin over the range searched, and both move smoothly with the region — so neither is a fragile answer that a small change in the brief would overturn. The disagreement between them is therefore structural rather than numerical noise, which is what makes it worth reporting rather than averaging away.

Where the model stops #

The blends are linear in the two parents’ page coordinates, which is what the library’s blend does and what Winkel did. A weighted geometric mean, or an interpolation of the two projections’ parameters rather than their outputs, is a different path through the same two endpoints and would have its own two optima.

The region is the world for the table and a thirty-degree cap for the overshoot, and the sweep is nine weights. A finer sweep moves the optima by less than a step and does not close the gap between them; a different region moves both, and moves them by different amounts, which is the region-dependence of every ranking here arriving in a new place.

And flexion is a root-mean-square over directions, from the rung that established the second derivative is not an invariant. A different summary of the same tensor — a maximum, or the component along a stated bearing — would give different optima again, and nothing here says how different.

The generalisation #

Averaging is exact at the order it is performed at, and says nothing about any higher one.

That is the whole rung in a sentence and it is not about projections. It is the reason a blend of two smooth functions can have a curvature neither has, the reason an interpolated table can oscillate, and the reason a mixture of two well-behaved models can be worse behaved than both — the mixture is exact in the value, and the derivatives were never part of the bargain.

The particular version here is worth carrying because the discipline has a strong habit of blending. Compromise projections are made by averaging, they are judged at first order, and the judgement is silent about a quantity the same maps also have.

Who found it, and when #

Winkel published his tripel in 1921 as the arithmetic mean of the equirectangular and the Aitoff, at a standard parallel he chose. That a compromise beats the mean of its parents’ errors at first order is implicit in every treatment of them and was made quantitative by the distortion measures of the 1970s and 1980s.

The second-order measures — flexion and skewness, in the form this anchor uses — are much younger, and were introduced to describe what a map does to a line rather than to a point. Applying them to the blend path is an obvious thing to do with two tools that exist, and this collection can find no record of anyone having done it.

What a designer has to decide first #

The two optima being half the axis apart is not a tuning difficulty; it is a statement that the blend parameter cannot be chosen without first deciding which order the map is for. There is no value that is good at both, and — since the two optima are that far apart — a value chosen for one is a long way down the other’s curve.

So the decision comes before the optimisation, and it is a decision about how the map will be read.

First-order judgements are made at a place. How big is this country, is this angle a right angle, does this shape look stretched. They are the judgements a reader makes standing still, comparing one part of the sheet with another, and they are what the angular and areal measures describe.

Second-order judgements are made along a line. Does this coast bend the way memory has it, is this road straight, does this great-circle route look plausible, does this boundary wobble. They involve the reader’s eye travelling, and they are what flexion and skewness describe.

The second kind is what a general-purpose map is mostly used for, which is the uncomfortable half of the finding. A reader of a world map is rarely measuring an angle and is continually following a coastline, a border or a route with their eye — and the whole compromise tradition has been tuned on the first-order measures, because those are the ones that were available when the tradition formed.

That is not an argument for retuning every compromise projection to the second-order optimum. It is an argument that the choice was never consciously made: a blend parameter picked to minimise angular deformation is answering a question about a reader who is standing still, and nobody asked whether that reader exists.

The honest design statement is a pair. Publish the blend parameter, the order it was optimised at, and the value of the other order’s measure there. Three numbers, all of which the designer computed anyway, and together they say what kind of map this is rather than merely how good it is.

Where the ladder goes next #

Nine rungs measure the second derivative of a map at a point, over a region, along a line and along a path between two maps. What none of them has is a reader: every quantity here is defined on the sheet, and the operations a second derivative actually spoils — laying a ruler, judging a bend, following a road — happen at a scale set by the eye and the hand rather than by the projection.