A family is not closed under averaging

Nine rungs of this anchor treat a family as a set: a construction with a parameter running through it, the members ordered along the parameter, the whole thing closing up at both ends. The conic is the whole family shows the cylindrical and the azimuthal as its two limits, reached exactly rather than approached.

This collection also averages projections. The average of two projections does it deliberately, and the site asserts that the Winkel tripel — on more atlas endpapers than any projection of the last century — is the average of an equirectangular and the Aitoff, to arithmetic noise.

Put the two together and there is a question nobody has asked: is the average of two members a member? It has to be answered before either rung means what it appears to, because a family that is closed under averaging is a very different object from one that is not.

What a distance between maps is #

Answering the question needs a distance, and positions will not do. A projection’s overall scale is free, its origin is free and its rotation is free, so two identical maps drawn at different sizes would come out arbitrarily far apart under any comparison of coordinates.

The invariants have none of that freedom. What survives a change of coordinates establishes which quantities describe the map rather than the drawing: a, b, the areal factor and ω, as against h and k, which depend on the graticule.

So the distance used here is the root-mean-square over the sphere of the difference in log(a/b) — the logarithm of the indicatrix’s axis ratio. It is zero between a map and itself, symmetric, positive between two different conics, and blind to everything a drawing is free to choose. All three properties are asserted before any of it is used.

The measurement #

Two members of the equal-area conic family, their average, and a search over the whole family for the member nearest it.

The residual is never zero and grows faster than the gap. Two neighbouring conics average to something 1.7 per cent of their own separation outside the family; two far apart average to something nearly a third of it outside.

And the nearest member is not the one at the average parameter. For the widest pair the parameter midpoint is 0.5 and the nearest conic is at 0.4505. Averaging two maps does not average their parameters, because a map is not linear in its parameter — which is report the map, not the parameters arriving as a statement about arithmetic rather than about reporting.

Why it happens, in one line #

An equal-area conic maps parallels to concentric circular arcs about the cone’s own apex, and the apex’s position depends on the cone constant.

Averaging two conics averages two families of arcs about two different centres. The result is a set of curves that are not circular arcs about anything, so no cone constant reproduces them, and the family cannot contain the average however its parameter is chosen.

That argument generalises immediately: a family is closed under averaging only if its defining construction is linear in the parameter, and almost no projection family is. The conic’s is a solve; the pseudocylindrical’s is a free function; the azimuthal’s is a radial profile.

What averaging is actually for #

The geometry says an average leaves the family. It does not say whether that is a gain or a loss, and the answer turns out to depend entirely on which two maps are averaged.

The criterion is Airy’s in Kavrayskiy’s form — the root-mean-square over the sphere of the two principal scales’ logarithms — which counts shape and size together, and which is what a general-purpose map is being asked to trade. Mean angular deformation alone ranks Mercator first and is therefore not a criterion for anything a compromise projection is for.

Averaging within a family is worthless, every time. The two parents are already making the same trade in the same way, differing only in where along the parameter they sit, so their average is a worse version of both — it inherits each one’s failures in the places the other did not fail.

Averaging across two families pays. The equirectangular’s worst regions and the Aitoff’s are different regions, so the average cancels part of each, and the result beats both by sixteen per cent.

That is why every compromise projection in the historical record is an average of two different constructions. Winkel’s is an equirectangular and an Aitoff. It is not a coincidence of taste; it is the only kind of average that buys anything.

The checks the measurement needs #

A residual that is never zero could be a residual that is never zero because the search is bad, so three things are asserted before the table is read.

The distance is a distance. Zero between a map and itself, symmetric between two, and positive between two different conics. A distance that failed the first would report a residual for a member of the family against itself, which is the failure mode a comparison of positions actually has.

The search finds the minimum. A coarse scan over the whole parameter range runs before the golden-section refinement, and the refinement is confined to the bracket the scan identifies — so a family with two local minima would be caught by the scan rather than missed by the search.

And the residual is smaller than the distance to the parameter midpoint. That is the check that the search found something: if the nearest member were the one at the average of the two parameters, the search would have been unnecessary and the family would be behaving linearly after all. It is not, on every pair — the member at the parameter midpoint is further from the average than the nearest member is, on all six.

Where the model stops #

One family and one condition. The equal-area conic is measured because its parameter is clean and its two limits are exact. A conformal or equidistant conic would give the same qualitative answer with different numbers, and a family whose construction is linear in its parameter would be closed — none of the site’s is, and the general claim is not made.

One criterion. The gain table is on the combined criterion; on mean angular deformation alone the Winkel tripel does not beat both parents, and on areal error alone the equal-area parents beat any average of them by definition. A gain is always against a criterion, which is why no projection is best without naming the purpose.

And the distance is one invariant. log(a/b) is the map’s shape distortion; a distance built from the areal factor instead would rank pairs differently. Both are invariant, so both are legitimate distances between maps, and the residual being positive does not depend on which is chosen.

What this changes about a parameter search #

The practical consequence lands on the landscape the search walks on and everything above it.

An optimisation over a family searches the parameter and returns the best member. This rung says the best map in the neighbourhood may not be a member at all: the averages of pairs of members are outside the family, and one of them may be better than every member of it.

For the conic family measured here, it is not — every within-family average is worse. So the search is safe on this family, and it is safe for a reason rather than by luck: the failure the family is trading is a single failure with a single parameter, and averaging two settings of one knob cannot beat the best setting.

Where the search is not safe is the case the compromise projections occupy: two families whose failures are in different places. There the search over either family alone misses the average by sixteen per cent, and no amount of refinement inside one family finds it.

Nearly closed, over a short interval #

The last figure is worth reading carefully, because it says what is true rather than only what is false.

Over a short interval the family is nearly closed. The residual for two conics a tenth apart is 1.7 per cent of their separation, the nearest member sits within 0.001 of the parameter midpoint, and for any practical purpose the average of two neighbouring conics is a conic.

That is what a smooth family has to do: locally, a curve in a space of maps looks like a straight line, so an average of two nearby points on it is close to the curve. The residual is second order in the separation, which is why it grows from 1.7 per cent to 31 per cent as the gap goes from 0.1 to 0.8 — a factor of eight in the gap and eighteen in the fraction.

So the failure is a large-separation failure, and the practical reading is a threshold rather than a prohibition. Averaging two members that are near neighbours is safe and pointless; averaging two that are far apart leaves the family and does not buy anything either.

The family is a curve, and the residual measures how sharply it bends #

The table’s last two columns hide a law, and it is worth extracting because it turns a list of six residuals into one number about the family.

Divide each residual by the square of the corresponding separation. The first four rows give 0.0992, 0.0992, 0.1000 and 0.1026 — four figures of agreement over a fourfold range, and then 0.143 and 0.222 for the two widest pairs. So

$$\text{residual} \;\approx\; 0.099\,d^2$$

with d the parents’ own separation, holding to a per cent up to d ≈ 0.7 and breaking above it.

That is exactly the shape a smooth curve makes. The midpoint of a chord across a circular arc of radius R sits a distance d²/8R from the arc — the sagitta — so a coefficient of 0.099 means the equal-area conic family is a curve of radius 1.26 in this space of maps. The family’s whole length is about 1.7 by the same measure, since the widest pair measured is 1.40 apart along the chord, so it bends through something like 1.35 radians end to end: a considerable arc, not a nearly straight one.

Reading it that way settles three things the table states separately.

Why the residual as a fraction of the separation is linear rather than constant. The fraction is 0.099d, so it doubles when the parents are twice as far apart, which is the 1.7, 3.3, 5.0, 7.0 per cent column read as arithmetic rather than as six measurements.

Why the nearest member drifts away from the parameter midpoint. The foot of the perpendicular from a chord’s midpoint lands at the arc’s own midpoint only if the curve is parameterised at constant speed. The conic family is not — the separation per unit of cone constant falls off above 0.4, which is visible in the “apart” column running 0.168, 0.336, 0.505, 0.677 and then falling behind its own linear extrapolation — so the nearest member is displaced towards the end of the family where the parameter moves the map faster. It is at 0.4505 for the widest pair, which is on the low side of 0.5, and the cone constant is indeed doing more work near the cylindrical limit.

And why the law breaks above d ≈ 0.7. A sagitta formula is the first term of an expansion in d/R, and at d = 1.03 against R = 1.26 the ratio is 0.8 — far outside where one term is enough. The widest pairs are not measuring a different phenomenon; they are measuring the same curve past the point where a quadratic describes it.

So the whole of the “not closed” finding is one geometric quantity: a family is closed under averaging exactly when its curve is straight, and the radius of curvature is how far from closed it is. For this family that radius is 1.26 and for a family defined linearly in its parameter it is infinite, which is the general claim the section above declines to make and this one can.

The generalisation #

A parameterisation is a coordinate system on a set of maps, and a set of maps has no reason to be a vector space.

The collection meets this repeatedly with different objects. Every equal-area map is every other one is the same observation about the equal-area condition: the set it defines is not a one-parameter family at all but a whole function’s worth of freedom. The family is a symmetry, not a shape is the same observation about what makes a family a family.

The transferable form is about optimisation rather than taxonomy. Whenever an answer is found by searching a parameter, the question is what the parameterisation cannot reach, and the cheapest probe is to average two members and measure how far outside the family the result lands. It costs two evaluations and a search, and on the family measured here it comes back at a third of the parents’ separation.

What a taxonomy is for, if not this #

The rung has a consequence for how the whole anchor’s vocabulary should be read.

A projection taxonomy — cylindrical, conic, azimuthal, pseudocylindrical, and the parameter inside each — is a way of naming maps that somebody has constructed. It is not a partition of the maps that exist, and it is not closed under any operation anybody performs on maps.

That is a weaker thing than it sounds like, and the site’s own machinery keeps demonstrating it. A blend is not in any family. A projection defined by a table is not in one either, and its interpolation scheme is part of its definition. Solving for the map instead of choosing it produces maps that satisfy a condition and belong to no named family at all.

So the taxonomy is a catalogue of constructions with names, which is exactly what a catalogue is for and is nothing more. Reading it as a classification of the possible is the mistake this rung measures the size of: the space between the named curves is where the compromise projections live, and they are the ones on the endpapers. Compromise projections is the rung that priced what they buy; this one says where they have to come from.

One practical note for anybody using the site’s own machinery. The distance defined here is available for any pair of maps, not only for members of a family, and it costs one distortion evaluation per sample point. So the question how far apart are these two projections now has an answer that is not a picture: two maps at a distance of 0.03 are the same map for any purpose a reader has, and two at 1.4 are not comparable at all.

Who found it, and when #

That a compromise projection is an average is Winkel’s own description of the tripel, published in 1921, and the arithmetic has been reproduced many times. Nothing about it is obscure.

What appears to be missing from the literature is the negative half — that averaging within a family is a waste of effort — which is worth having because it says where to look. A cartographer trying to improve on a conic by blending two conics is doing something the geometry rules out in advance; one blending a conic with something structurally different is doing the thing that has historically worked.

The distance between maps used here has cousins in the literature — the various “projection similarity” measures used for identification, which a map does not say what it is uses for a different purpose — and they are usually built on positions after a best fit rather than on invariants. Building it on the indicatrix instead is what makes the residual above a property of the maps rather than of the fitting.

Where the ladder goes next #

Ten rungs price a family’s members, its parameter, its limits, its condition, its symmetry and now its closure. What none of them prices is the set that all the families sit inside: every projection here is a smooth map from the sphere to the plane, that set is enormous, and the families are a handful of curves through it. How much of the set the named families cover — and whether the best map for a stated purpose is on any of those curves — is the question the anchor has been circling for ten rungs without asking.