The second derivative is not an invariant

The very first thing this collection established about distortion is that some of the numbers describing it are real and some are bookkeeping. The scale along the meridian and the scale along the parallel depend on how the sphere was parameterised; the two principal scale factors, the areal factor and the maximum angular deformation do not. What survives a change of coordinates is the essay that sorted them, and everything since has quoted the survivors.

The flexion ladder has produced five rungs of second-order quantities and has never been asked the same question.

Tissot stops at the first derivative opened this ladder by pointing out that a map can be conformal at a point and still bend every geodesic through it. Six rungs later, the quantities that measure the bending have never been sorted the way the first-order ones were on the day they were introduced.

The transformations that matter are of the page #

There is an important difference between this question and the first-order one, and it is worth stating before any numbers.

At first order the transformation in question was a change of coordinates on the sphere — measure latitude differently, and h and k change while a, b, the areal factor and ω do not. That is a question about the parameterisation of the domain.

At second order the interesting transformations are of the page, because a printed map undergoes them constantly and nobody records that it has:

• a similarity — a rotation and a uniform magnification. Every reprinting, every reduction to fit a column, every rotation to fit a page.
• an anisotropic scaling — one axis stretched. A non-square pixel, a printer with different horizontal and vertical resolution, a plot rendered at whatever aspect the window happened to have.
• a shear — a scanner whose transport is not quite square, a photograph taken off-axis, a reprojection that happens to be affine.

What a careless copy hides established that an affine fit absorbs the whole difference between the cylindrical equal-area projections, so an affine map of the page is not an exotic thing to worry about — it is the transformation that makes two named projections indistinguishable.

It is also the transformation this collection has already found doing damage elsewhere: an affine fit removes the whole difference between two named projections, and a reprojection between two conformal maps is a similarity everywhere while a reprojection between anything else is not.

The control, which has to pass first #

Flexion and skewness are turning per unit of arc on the ground, not per unit of arc on the page. So they are already dimensionless, and a uniform magnification cannot touch them.

The measurement confirms it to every printed figure: flexion 0.4338 before and 0.4338 after, skewness 0.2378 and 0.2378, ω 10.9732 and 10.9732, anisotropy 1.2114 and 1.2114. A rotation through 25° and a magnification by 1.6 changed nothing.

The last column of that figure is the same quantity measured per unit of page arc, which does carry a unit, and it moves by exactly the magnification — 0.4775 to 0.7639, a factor of 1.600000. Keeping the wrong version beside the right one is what makes the right one legible: the pair says which of the two a published number is, and the answer for every score in the literature is the first.

What breaks it #

An anisotropic scaling by 1.6 raises the flexion from 0.4338 to 0.4798 — 10.6 per cent — and raises the maximum angular deformation from 10.97° to 19.00°.

A shear of 0.5 raises the flexion to 0.5874 — 35 per cent — and ω to 26.41°.

Both of those are entirely expected once stated: an affine map of the plane is not a similarity, it takes circles to ellipses, and every quantity that describes the shape of an image is going to move. The first-order quantities move too, and visibly: the anisotropy goes from 1.211 to 1.392 and to 1.592.

So the second-order quantities are not worse behaved than the first-order ones. They are exactly as badly behaved, and this is the first rung to say so, because the first-order failure is already understood and priced — a reader who is told the anisotropy of a printed map knows to ask what the printing did to it — while a second-order score is quoted as though it were a property of the projection.

The stronger question: does the ranking move #

One map’s number changing is not automatically a problem. A second-order score is used to rank maps, and a ranking survives any transformation applied monotonically to every score. If the same change of page were applied to every candidate — which is what happens when eight world maps are printed in one atlas, at one aspect ratio — the ranking might be untouched.

It is not untouched, and the threshold is low.

A stretch of 1.6 in one axis is not a subtle thing to look at — but it is exactly the sort of difference between two published renderings of the same world map that nobody records, and it moves half the ranking.

A shear does worse. Applied at k = 3 to all eight candidates, the rank correlation with the original ordering is −0.07, which is no relationship at all, and Mercator moves from fifth of the eight to eighth.

Why the second order is more fragile than it looks #

The mechanism is worth naming because it explains why the threshold is where it is.

Flexion at a point is the rate at which the image of a geodesic turns, maximised over directions. An affine map of the plane does two things to that: it changes the curvature of every image curve, and it changes which direction is the worst one. The first is a smooth distortion of a number; the second is a discontinuous relabelling of what is being measured, and it is what makes the ranking jump rather than drift.

The same is true of ω, which is also a maximum over directions, and which is also not affinely invariant. The difference is that ω’s dependence on the anisotropy is well known and the flexion’s is not reported anywhere.

What a score should say instead #

There is a version of the quantity that survives an affine map, and it is available from the machinery already here: measure flexion in the ground frame carried through the map’s own first-order part. That is what the ratio flexion / a is doing implicitly, and it is not enough — the first-order part is a full 2 × 2 matrix and dividing by one of its singular values removes only the isotropic component of the change.

The honest general statement is that a second-order score is a joint property of a projection and the affine class of the page it is drawn on, and the useful way to quote one is with the page normalised: a stated aspect ratio and a stated orientation, exactly as a representative fraction is only meaningful with a stated point. Nothing in the literature does this, and until this measurement there was no reason to suppose it mattered.

The second derivative has its own ranking is the rung that produced a ranking and compared it with the first-order one, finding Spearman 0.45 between bending and stretching. That number is itself a statement about one drawing of each of the eight maps, and now has to be read that way.

The one case where it does not matter #

There is a class of maps for which none of this bites, and naming it sharpens the rest.

If a projection is compared with itself under a change of page, the transformation is a similarity whenever the two drawings differ only in size and orientation — which is the case for any single map reproduced at different scales. Every printed reduction of a world map, every thumbnail, every zoom of a raster copy: all similarities, all leaving flexion exactly where it was. That is why the fragility has never been noticed, and it is a real exemption rather than a lucky one.

What breaks it is comparison between drawings that were prepared independently. Two atlases showing the same eight projections at two different page aspect ratios will disagree about the second-order ranking of those eight, by up to four places out of eight, with neither of them wrong. Two grids over the same ground makes the same kind of point about coordinates: two correct descriptions of the same thing need not agree on any derived quantity unless the derivation was invariant.

What was computed, and how #

A page transformation is applied by wrapping a library projection so that its forward map is composed with a fixed 2 × 2 matrix, leaving the metric, the visibility rule and the limit untouched. Every quantity is then measured by the same code that measures anything else — flexion and skewness by finite differences along geodesics fired in seventy-two directions, the first-order quantities by the Jacobian.

The regional score is a weighted root mean square of the flexion over Europe, sampled on a grid with the area element as the weight, which is the same score the second derivative over a region uses.

The assertions require the similarity to leave flexion alone to a part in ten thousand and to multiply the per-page-arc version by exactly 1.6, so a bug that ignored the transformation entirely would pass the first and fail the second. They also require the anisotropic scaling and the shear each to move flexion by more than five per cent, which is the refusal.

The rank correlation is Spearman’s, computed on the positions rather than on the values, so that a monotone change in every score would leave it at 1 by construction — which is the whole point of using it.

What a paper should report #

The repair is not to abandon the measure. Flexion and skewness are real quantities that describe something the first-order apparatus cannot see, and this rung does not weaken that.

The repair is to report the page. A second-order score should arrive with the aspect ratio and the orientation the projection was drawn at, exactly as a scale factor arrives with the point it was measured at and a distortion criterion arrives with the region it was integrated over. Every other quantity in this collection carries its own context and this one has been quoted without any.

That is a small change to a convention and it makes a published number reproducible. Without it, two people computing the flexion of the same projection over the same region will agree only if they happened to normalise the page the same way — and nothing in the literature tells them how.

Where the model stops #

The transformations here are global and linear. A real reproduction is neither: a scan has a slowly varying distortion across the sheet, a photograph has perspective, and both are locally affine and globally not. The local statement is the one measured, and it is the one that matters for a pointwise quantity, but a score integrated over a region under a varying affine map is not what is computed here.

Only flexion drives the ranking figure. Skewness behaves similarly at a point — it moves under a shear by four per cent against flexion’s thirty-five — and its ranking has not been re-run, because the regional score for eight maps under six transformations is the most expensive measurement in this ladder and doubling it buys a second number rather than a second finding.

And nothing here says which page is the right one. There is no privileged aspect ratio for a world map; the choice is a design decision made by whoever drew it. What the measurement establishes is that the decision is not free, which is a different thing from knowing how to make it.

It is also the ordinary way a measure becomes standard. A paper introduces a quantity, computes it for a set of examples in one convention, and the table is quoted afterwards without the convention — which is invisible until somebody applies the quantity to a case the convention does not cover. Here that case is any map somebody else drew.

Who found it, and when #

Flexion and skewness in this form are Goldberg and Gott’s, from their 2007 paper on flexion and skewness in map projections, which introduced both as measures the classical apparatus was missing and ranked a set of world maps by them. The paper computes its quantities from the projections’ formulae, which fixes the page implicitly — every candidate is drawn in its own conventional normalisation and nobody thought to ask what happens if it is not.

That is the ordinary way an invariance question gets missed. Tissot’s own quantities were sorted into invariant and not because the classical treatment worked in an explicit change of coordinates and had to; a modern measure computed from formulae in a fixed convention never meets the transformation that would ask the question.

Three ways to publish a measure that is not invariant #

The finding is that a quantity was introduced with a normalisation nobody stated, which is a specific defect with three general remedies. They are not equally good and all three are better than what is done.

State the normalisation. The weakest option and the cheapest: say which page the numbers were computed on — the aspect, the scaling, the units, whatever fixes the drawing — so that a reader recomputing the quantity for a new map knows what to match. A table with a stated convention is reproducible even if the quantity itself is convention-dependent.

Report a ratio that is invariant under it. The stronger option, when one exists: divide the quantity by something transforming the same way, so the result is a pure number independent of the drawing. This is what the classical apparatus does — the angular deformation and the areal factor are invariants precisely because they are ratios of things that scale together — and it is why Tissot’s quantities survived being quoted without their conventions for a century.

Report the ranking under two normalisations. The pragmatic option when no invariant ratio is available: compute the comparison twice, on two different drawings, and say whether the order changes. A ranking stable across two arbitrary conventions is evidence, though not proof, that the comparison is about the maps; a ranking that moves has been measured about the drawings, and the reader learns that immediately rather than in someone else’s later paper.

The three are in increasing order of what they establish and are all one extra pass, which is the argument for doing at least the first. What was published instead is a table with no convention, which supports none of the three and is the reason this rung had to be written at all.

And the diagnosis generalises past this measure. Any quantity computed from a projection’s formulae in each projection’s own conventional normalisation has the same exposure, and the exposure is invisible while every candidate is drawn conventionally. It surfaces the first time somebody applies the measure to a map they did not draw — which is the use every such measure is proposed for.

Where the ladder goes next #

Six rungs have established what the second derivative is, that it is independent of the first, that it ranks maps differently, that it has a regional form, that a local polynomial model has an order, and now that it is a property of a drawing rather than of a projection. The remaining question is the one that would make it useful: what a reader actually sees, which is neither the pointwise rate nor the regional mean but the departure of a drawn line from a drawn geodesic over a finite length — and which the site has measured once, in passing, as the bow of a geodesic.