An average of ellipses is not an ellipse

An atlas that prints a distortion figure for a projection is printing an average. There is no such thing as the distortion of the Robinson projection; there is a distortion at every point, and a table entry is what happened when a few thousand of them were reduced to one number. Distortion over a region is the essay that establishes this, and it establishes the part everybody argues about: the reduction requires a weighting, the weighting is a choice, and the ranking of projections moves when the choice moves.

That essay averaged numbers. Airy’s criterion averages a squared departure of the two principal scales from one; Kavrayskiy’s averages the squares of their logarithms; the mean angular deformation averages an angle. Each of those is a scalar computed at a point and then integrated, and integrating a scalar is uncontroversial once the weighting is settled.

The thing being summarised is not a scalar. It is Tissot’s indicatrix, and an indicatrix is a matrix — the metric tensor G = AᵀA of the map from the local ground frame to the page, whose two eigenvalues are a² and b². Symmetric positive-definite matrices do not form a vector space. They form a cone, and a cone has more than one defensible notion of an average.

Four averages of the same ellipses #

The four are worth naming, because three of them are in use and one of them is what everybody actually does.

The arithmetic mean of the tensors. Add the matrices, divide by the count. It is the first thing anyone writes, it produces a perfectly good positive-definite matrix, and it is a genuine average in the sense that it minimises the sum of squared Frobenius distances.

The log-Euclidean mean. Take the matrix logarithm of each tensor, average those, exponentiate the result. This is the natural mean when the quantity being averaged is a ratio rather than a difference — and every scale factor on this site is a ratio. A scale factor of 2 and a scale factor of ½ average to 1 under this rule and to 1.25 under the previous one.

The affine-invariant, or Karcher, mean. The point that minimises the sum of squared distances in the cone’s own Riemannian metric, found by iteration. It is the mean that does not care what linear change of ground coordinates was used to write the tensors down, which is exactly the property this site demands of anything that describes a map.

The mean of the axes. Average a, average b, and draw the ellipse with those semi-axes. This is not an average of matrices at all — it discards the orientations before combining anything — and it is what a published “average distortion” almost always is.

The finding is in the determinants #

det G is the square of the areal factor. An equal-area projection has det G = 1 at every point of every region, exactly, by construction, and that is the one property such a projection exists to have.

That is Minkowski’s determinant inequality doing its work: for positive-definite matrices, det((A+B)/2)^{1/2} ≥ (det A^{1/2} + det B^{1/2})/2, with equality only when the two are proportional. So the arithmetic mean of a set of unit-determinant tensors has determinant strictly greater than one unless every one of them is identical.

The log-Euclidean mean has determinant exactly one, and the reason is one line: the determinant of a matrix exponential is the exponential of a trace, the trace of a mean is the mean of the traces, and the trace of log G is log det G, which is zero for every input. So the mean of the logs has zero trace and its exponential has unit determinant, whatever the tensors were.

The summary destroys the property the projection was chosen for. Somebody comparing equal-area projections by averaging their indicatrices and reading off the areal factor of the average would find that none of them is equal-area, that they differ from each other by tens of per cent in a quantity that is identically one, and that the ordering of that non-existent quantity is stable and reproducible. Every part of that is an artefact of an arithmetic convention nobody wrote down.

The deformation of the average is not the average of the deformation #

The areal factor is the clean case because the true answer is known in advance. The angular deformation has no such anchor, and its behaviour is stranger.

The alignment number is the explanation and it separates the population completely. An ellipse axis is a line rather than a direction, so the average orientation of a set of them is taken on the doubled angle; the resultant length comes back at one when every ellipse points the same way and at zero when they are spread evenly.

The three cylindrical projections in that figure — Miller, the plate carrée and the Lambert cylindrical — return an alignment of exactly 1.0000. On a cylindrical map the meridian and the parallel are the principal directions everywhere, and with the equator as standard parallel the parallel is the stretched one at every latitude, so all sixteen hundred ellipses have their long axis east–west. Nothing cancels. The average is more deformed than the average deformation, by eleven degrees on the Lambert cylindrical.

The seven others return alignments between 0.227 and 0.567. On a pseudocylindrical map the principal directions swing round with longitude, ellipses at one edge lie across ellipses at the other, and the tensor sum partly cancels. The average is less deformed than the average deformation, by up to twenty-nine degrees.

So the sign of the discrepancy is not a property of the projection’s distortion at all. It is a property of how that distortion is arranged, and the two are independent: Gall–Peters and the Lambert cylindrical are both equal-area cylindricals with the same family of ellipses, and they land on opposite sides of the diagonal because one has its standard parallels at 45° and the other at the equator.

And the ranking moves #

The practical question is whether any of this changes an answer. It does.

Eight of eight move. The plate carrée is seventh under the log-Euclidean mean and second under the mean of the axes. Eckert IV is first under three of the four and fifth under the fourth. Gall–Peters is second under the arithmetic mean and third under the log-Euclidean.

This is the region-dependence result arriving from a direction that essay could not see. There, the ranking moved because the region moved and because the criterion moved, and both of those are visible choices that a careful table states. Here the region is fixed, the criterion is fixed — maximum angular deformation, on every column — and the ranking moves anyway, because of a step between the samples and the summary that no published table mentions and that most authors would not describe as a choice at all.

What decides how much it matters #

The gap between the four averages is not a constant of the projection. It grows with how far apart the region’s own indicatrices are.

That is the practical rule and it is a reassuring one. A national atlas can average distortion carelessly and a world atlas cannot. Over a country a few hundred kilometres across, every indicatrix is nearly the same ellipse, all four means agree to a part in a thousand, and the distinction drawn in this essay is a pedantry. Over a hemisphere it is a factor.

It also explains why the problem has stayed invisible. Regional distortion analysis is where the numbers get used — choosing a projection for a country is a live question and the answer is checked against practice — while world-map distortion figures are quoted in introductions and compared to nothing.

What an average ellipse is a claim about #

There is a question underneath all of this that the arithmetic does not answer: what is the average indicatrix supposed to be an average of?

Three readings are in circulation and they want different means.

A typical value. The ellipse a reader should expect at a place picked at random in the region. That is a statement about the distribution of the tensors, and the mean that answers it is the one that behaves like the middle of the data — which for ratio-valued quantities is the log-Euclidean or the Karcher mean, in the same way that the typical value of a set of scale factors is their geometric mean and not their arithmetic one.

A total. The distortion the region suffers in aggregate, the way a total area or a total error would be. That reading genuinely wants sums, and the arithmetic mean of the tensors is the right object for it — but then the answer should be reported as a sum with units of area, not drawn as an ellipse, because the ellipse invites the first reading.

A representative map. The single linear map that best stands in for the projection over the region, in the sense of a least-squares fit. That is a fourth question again, its answer depends on what is being fitted, and it is the one an atlas caption most nearly implies and least often means.

The four means in this essay are answers to different questions and there is no way to tell from a table which question was asked. That is the whole of the finding, and it is the same shape as the shortfall rung three records about the weighting: a step in the reduction that looks like arithmetic and is a decision.

The one case where all four agree #

They agree exactly when every tensor in the set is identical, which happens on a real map exactly once: a conformal projection with constant scale, which is a similarity, which does not exist on the sphere.

They agree to any stated tolerance when the tensors are close enough together, and the spread figure above is the measurement of what “close enough” means. Below an axis-ratio spread of about 1.2 — a region a few hundred kilometres across — all four means agree to better than a part in a thousand and the distinction is academic.

There is also a case worth stating because it looks like an exception and is not. On a conformal projection every indicatrix is a circle, so every tensor is a multiple of the identity, so all four means are circles and their orientations do not matter. But they are still four different circles: the arithmetic mean returns the root-mean-square scale factor, the log-Euclidean and Karcher means return the geometric mean, and the mean of the axes returns the arithmetic mean of the scale factor. Over the whole sphere on a Mercator those three differ by more than a factor of two, and every one of them is a legitimate answer to a different question about the same map.

The arithmetic mean is never the conservative one #

The four means are presented as four defensible answers, and there is one thing to be said in favour of preferring any of the other three: the arithmetic mean of the tensors is biased in a known direction, always, in both of the families a cartographer cares about.

On an equal-area map it over-states the area. That is Minkowski’s inequality, and it is strict unless every tensor in the set is proportional to every other, so there is no region and no equal-area projection for which the arithmetic mean returns one. The bars run from 0.52 per cent to 10.95 and every one of them is positive.

On a conformal map it over-states the scale. Every tensor is then a multiple of the identity, so the three matrix means reduce to three means of a positive number: the arithmetic tensor mean returns the root-mean-square of k, the log-Euclidean and Karcher means return its geometric mean, and the mean of the axes returns its arithmetic mean. Those are ordered — geometric ≤ arithmetic ≤ root-mean-square, for every set of numbers whatever — so the log-Euclidean answer is the smallest of the three and the arithmetic tensor mean the largest, and the ordering does not depend on the projection or the region.

And on an aligned map it over-states the deformation. The three cylindrical projections whose ellipses all point the same way come back with the average more deformed than the average deformation, by up to eleven degrees; only the maps whose principal directions swing round get the cancellation that pulls the other way.

Put together, the mean that is easiest to compute is the one that reports the most distortion in every case where the answer is checkable. That is not an argument that it is wrong — it is the right object for the total reading, and a total should be large — but it is an argument against reaching for it by default, since a reader shown an average ellipse takes it as a typical one, and the typical value of a set of ratios is their geometric mean.

The one-line rule is therefore: average logarithms. It gives the geometric mean for a conformal map, unit determinant for an equal-area one, and the middle of the data for anything else — three properties that no other candidate has together, obtained by taking the logarithm of a matrix rather than of a number.

Where the model stops #

The four means are not an exhaustive list. There are others in use for tensor data — the harmonic mean, the Cholesky-Euclidean mean, the root-Euclidean mean — and each would give a fifth and sixth ellipse. Nothing here says which is right, because that depends on what the average is being asked for, and the point of the essay is that the question has to be asked at all.

The weighting is the previous rung’s problem and is not solved here. Every mean above uses the same area weighting over the same sample points, so the four columns differ only in how the tensors were combined. Change the weighting and all four move together, which is the effect rung three measured and which is orthogonal to this one.

The Karcher mean is computed by iteration and its convergence is asserted rather than proved. The fixed point exists and is unique for positive-definite data — that is a theorem — and the iteration used here converges in fewer than a dozen steps on every set of tensors a real projection produces. A pathological input could take longer, and the code stops at sixty steps rather than at a tolerance it cannot guarantee.

The equal-area demonstration depends on the library being exactly equal-area. It is: the areal factor of every projection in the figure is one to a part in 10¹² at every sampled point, which the site’s gate requires and rejects on, so the inflation in the bars is entirely the average’s own.

Who found it, and when #

None of the mathematics is new or contested. Minkowski’s determinant inequality is from 1896. The log-Euclidean mean was named by Arsigny, Fillard, Pennec and Ayache in 2006 and the affine-invariant mean is older than that, both of them arising in diffusion tensor imaging — a field which averages positive-definite matrices constantly, discovered the same problem, and settled it before anyone needed it here.

What is new is applying it to Tissot. The indicatrix has been the standard object of distortion analysis since 1859 and averaging one over a region has been standard since Airy proposed his criterion in 1861, and the averaging has always been done on scalars derived from the ellipse rather than on the ellipse. That is a defensible engineering choice; it is just not the same object, and the practice does not distinguish them.

Where the ladder goes next #

The indicatrix ladder has now taken the object apart at a point, over a region, at a point where it does not exist, and as a summary statistic. What it still treats as given is that the indicatrix describes something a reader can use — that quoting a and b at a place tells someone what will happen to a shape there.

The flexion ladder has already found the size at which that stops being true, and the gradient ladder has found a quantity the indicatrix describes backwards. What neither has asked is what a reader is entitled to do with an average — whether the ellipse in the atlas is a prediction about any place at all, or a bookkeeping entry that happens to be shaped like one.