The size at which the second derivative arrives

This ladder opened by observing that Tissot’s indicatrix stops at the first derivative and has spent six essays on what the second one does: how it is defined, how it varies with direction, how it behaves over a region, whether it survives a change of page, and how it relates to the polynomial a georeferencing tool fits.

What none of those essays says is when it matters. Flexion is quoted per unit of arc, which is a rate; a reader is entitled to ask at what size the rate has accumulated into something worth caring about, and the honest answer requires a figure that has an extent.

The decomposition #

Take a point, and geodesic normal coordinates around it: a displacement is a bearing and an arc length, and the position it names is found by walking that far along that great circle. Those coordinates are the right ones and not an arbitrary choice — in them the sphere’s own metric is the identity to second order, so any quadratic term that survives belongs to the projection and not to the coordinates it was written in.

The composed map from those coordinates to the page has a Taylor expansion. Its linear part is exactly the matrix Tissot’s indicatrix is the image of the unit circle under. Its quadratic part is six numbers, and they are what flexion and skewness are built from — resolved into a tangent and a normal component rather than into page axes, which is what makes them invariant.

Project a circle of radius ρ and compare where its points land with two predictions:

• first order, which uses the linear part alone;
• second order, which adds the quadratic terms.

Each departure is quoted as a fraction of the figure’s own drawn size, because an absolute departure on a page whose units are arbitrary says nothing.

What the errors do #

The exponents are the reason to trust the rest. A model that was merely smaller than another could be smaller for any number of accidental reasons; a model that converges one order faster is doing what its construction says it does. Over six sizes spanning a factor of thirty-two, Mercator gives 1.08 and 2.08, Mollweide 1.01 and 2.05, and the Albers conic 1.02 and 2.05.

The relative error is a first power rather than the second a reader might expect because the absolute departure is quadratic and the figure’s own size is linear. Quoting the absolute number would say that a small figure is placed well, which is trivially true and tells nobody anything about whether the picture is right.

The size at which each model runs out #

The numbers are the point of this rung, so they are worth reading slowly. At 20°E 45°N, to one part in a thousand:

• Mercator: the indicatrix is good to 12.7 km, the second-order model to 279 km — a factor of 21.9;
• the Lambert cylindrical equal-area: 11.4 km and 290 km;
• the Albers conic: 52.0 km and 423 km;
• the stereographic: 28.3 km and 543 km;
• the gnomonic: 5.7 km and 157 km.

Thirteen kilometres is the scale of a city. The claim that Tissot’s ellipse describes what a projection does to a shape is therefore true, at the accuracy anybody would use the word describes for, over an area smaller than a metropolitan borough — and every atlas figure that draws indicatrices the size of Greenland is drawing a first-order model outside its own domain by four orders of magnitude.

That is not a complaint about the figures, which are diagrams of a rate rather than predictions. It is a measurement of the gap between what the diagram shows and what it is taken to mean.

Which projection lasts longest, and why it is not the gentlest #

The ordering in that list is not the ordering of distortion. The gnomonic is the worst-behaved projection in the library by every first-order measure and it has the shortest usable radius; so far so expected. But the Albers conic — a middling projection by any first-order score — carries its indicatrix four times further than Mercator does, and Mercator is conformal.

The reason is that the usable radius is set by the second derivative and the ranking is by the first. A conformal map has no angular deformation at all, so its indicatrix is a circle and its first-order model is as simple as a model can be; that says nothing about how quickly its scale factor changes, which is what the quadratic term measures. Mercator’s scale factor is sec φ, whose derivative at 45° is large.

That is the same lesson the second-order ranking reached from the other end: the two orders rank projections differently, and a map chosen for one is not chosen for the other.

What sets the radius, direction by direction #

The usable radius is a single number for a figure that reaches every way at once, so it is worth asking which direction decides it. The answer is the one where the quadratic form is largest, and the roses above say that this direction is not the same for every projection and is not aligned with anything a reader would guess from the first-order picture.

That has a practical consequence for anybody choosing a projection for a long thin region — a coastline, a corridor, a national grid strip. Such a region’s extent is large in one direction and small in the other, so the size at which its second order arrives is set by the quadratic form in the direction the region runs, and a projection whose worst direction lies across the region rather than along it will hold a first-order description for far longer than the circle test suggests.

The circle test is the conservative one, which is why it is the one quoted. A region-shaped test would give a larger radius for every projection and a different ordering, and it would be a measurement of the region as much as of the map — which is exactly the confusion ranking over a region exists to keep out of the first-order numbers.

The refusal: a projection with no second derivative #

At a radius of 2° the analytic projections gain a factor of 16 to 55 from the second-order term. The tabulated projection gains 0.074: its second-order prediction is 13.5 times worse than its first-order one.

This is the check that separates a measurement of a map from a measurement of arithmetic. A projection interpolated between published rows has second differences, and they are perfectly real numbers; they describe where the interpolating polynomial bends between the tabulated latitudes, which has nothing to do with the surface being mapped. Feeding them into a Taylor expansion of the map produces a prediction that is confidently wrong.

It also settles a question the ladder raised and left open. One projection in the library has no second derivative at all — and what “no second derivative” costs, quantitatively, is that the whole apparatus of this ladder does not apply to it, and applying it anyway makes things worse rather than leaving them unchanged.

What was computed, and how #

The Jacobian and the six quadratic coefficients are central differences of the composed map at a stated step, taken in geodesic normal coordinates rather than in longitude and latitude. The step is the one this ladder’s step study settled on, sitting between the truncation error that falls as its square and the cancellation error that rises as its inverse square.

The truth is the projection’s own forward map applied to points found by walking the exact great circle — the plane through the centre containing the point and the tangent direction, rotated by the arc length — rather than by stepping an integration. An integrated path carries its own truncation error, and this is the one place it could not be tolerated: the quantity being measured is the difference between the truth and a quadratic model, and a truth with a quadratic error in it would be measuring the integrator.

The usable radius is found by bisection on the departure itself rather than read off the fitted power. The fit is a check on the mechanism; the radius is a measurement, and reading it off a fitted line would make it a consequence of the fit.

One caution is recorded in the machinery rather than papered over. For the tabulated projection the bisection has nothing to find — its second-order error never falls below a thousandth at any size in the bracket — so both of its radii come back sitting on the search’s floor. That is a fact about the bracket and not about the map, which is why the tabulated case is quoted as a gain at a stated size instead.

What the numbers mean for the figures this site draws #

This collection draws indicatrices on world maps constantly, and it is worth saying explicitly what the measurement above implies about them, because the answer is not that they are wrong.

An indicatrix drawn at 13 pixels across on a 660-pixel world map is a symbol. Its size is chosen so that a reader can see its shape, and its shape is the limiting shape, which is exact. Nothing about it claims that a region of that size on the ground deforms that way — and at the scale such a figure is drawn, a region 13 pixels across is several hundred kilometres, where the first-order model is out by several per cent.

So the figure is a correct diagram of a rate drawn at a size where the rate does not describe anything. That is a normal state of affairs for a diagram — a vector field’s arrows are drawn at a length nothing travels — and it becomes a defect only when the diagram is read as a prediction. The two readings differ by the factor this rung measures, and the collection’s own convention of stating a, b, the areal factor and ω in figures beside the ellipse exists precisely so that the numbers rather than the picture carry the claim.

The one place it does bite is the reader’s intuition about shape. An indicatrix that is circular invites the conclusion that a country drawn there keeps its shape, and it does not — a country is thousands of kilometres across and the first-order model expired at thirteen.

Where the model stops #

One point, one latitude. Every number above is at 20°E 45°N. The usable radii move with position: at 65°N the same two projections give 5.9 km and 151 km for Mercator, and 11.8 km and 196 km for Albers, so the whole ladder shifts down as the scale factor’s variation grows. The ordering survives; the numbers do not transfer.

A circle is one shape. The figure used is a geodesic circle, which is the natural test object because it is what the indicatrix is a statement about. A long thin region would give a different answer, and a smaller one, because the departure is governed by the extent along the direction the quadratic term is largest in.

One part in a thousand is a choice. It is a plausible standard for a drawn figure — half a millimetre on a page — and nothing in the measurement depends on it. A tolerance of one part in a hundred multiplies the first-order radius by about ten and the second-order one by about three, because the errors are of different orders.

Third order is not measured. The residual after the quadratic term includes everything above it, so the second-order curve’s exponent of 2 is a statement that the next term dominates the residual. What the third derivative would buy is a further order, and this ladder has never argued that the expansion should be carried further — only that stopping at the first is a decision with a size attached.

The generalisation #

The rule is a pair of numbers per projection and it can be stated as one sentence: an indicatrix is a description of a map over a town, and a second-order model is a description over a county.

Anything larger needs the projection’s own formula, which is available and exact and is what every piece of software actually uses. So the practical consequence is not that anybody should be computing quadratic terms — it is that the understanding a reader carries from a Tissot diagram is quantitatively good over a range some four orders of magnitude smaller than the map it is drawn on.

And that is the honest reading of what this ladder has been doing. Flexion is not a correction anybody applies. It is the measurement that says how far the first-order picture can be trusted, and this rung is where it finally produces a distance.

Who found it, and when #

Tissot’s 1881 treatment is explicitly infinitesimal, and he says so; the ellipse is defined as a limit and the whole of the classical theory is about rates.

Goldberg and Gott introduced flexion and skewness as second-order measures in 2007, and gave them the units of turning per unit of arc — which is a rate again, and left the question of accumulated size where it had been. Their motivation was comparative rather than predictive: to rank projections on a criterion the first order could not see.

The size at which a local model runs out is a standard question in numerical analysis and an unusual one in cartography, because cartography almost never uses the local model for anything. It is used for understanding, and understanding does not come with an error bar unless somebody computes one.

A model used for understanding is still used for prediction #

The remark that cartography uses the local model for understanding rather than for computation is true and it understates what is at stake, because understanding is a kind of prediction and it is the kind nobody checks.

A reader who has understood a claim believes things about a neighbourhood. Told that a projection is conformal at a point, or that the indicatrix there is a circle, a reader does not stop at the point — they form an expectation about the region around it, because a statement about a single point is of no use to anybody and nobody treats it as one.

That expectation is a local model applied over a finite size, and the size is exactly what this rung measures. It is not written down, it is not taught, and the reader has no way to guess it: nothing about the phrase conformal at this point suggests whether the neighbourhood in question is a kilometre or a thousand.

So the radius matters even where no software uses it. A reader forming a belief is running the same first-order model a calculation would, with the same validity limit, and without the calculation’s opportunity to be checked against something.

Which is the argument for computing it in a collection like this one. The numbers here are not for a pipeline; they are the missing half of a sentence that is otherwise unfalsifiable. The map is conformal here becomes the map is conformal here, and behaves that way out to about this far, and the second is a statement a reader can act on and be wrong about.

Where the ladder goes next #

Seven rungs of this ladder have treated the map as exactly known and asked what its derivatives do. The natural next question is what happens when the map itself is uncertain — when the transformation between two systems is fitted from control points rather than given by a formula, so its first and second derivatives carry the fit’s own noise.

That is a different kind of second derivative, and it belongs beside a coordinate that is a number with a width rather than here.