Cartographic Projection
every map is wrong, and by exactly how much
Measuring distortion

The second derivative over a region

Flexion has been measured at points and over the whole sphere, and never over a region — which is the only unit anybody chooses a projection for. Doing it finds that the second-order criterion moves the winner in one region of four, and that one projection in the library has no second derivative at all.
15 min read 8 figures Measured, not namedWhat survives a change of coordinatesPurpose before property

The second derivative has its own ranking ranks eight projections by their flexion and skewness over the whole sphere and finds a rank correlation of 0.69 with the first-order ranking of the same eight. The conclusion there is that the two orders measure different derivatives of the same maps and there is no reason for them to agree.

A whole-sphere ranking is the wrong shape for a decision, and distortion over a region says why: a projection is never chosen for the sphere. It is chosen for a country, a continent or a corridor, and a ranking that integrates over the whole world is a ranking about a map nobody is making.

So the question this rung asks is whether the second order’s disagreement with the first survives being asked about a region.

Which projection wins, by the first derivative and by the second. Each column is a region, with the projections listed in the order Kavrayskiy's first-order criterion puts them in and the figure at the right of each row giving that projection's rank under the second-order criterion — flexion and skewness aggregated the same way. The rank correlations are Europe 0.83, the conterminous United States 0.81, the tropics 0.90, a cap of 30° radius 0.88, so the two orders agree broadly and disagree in detail. Where it matters is the winner: over a cap of 30° radius the choice moves from Albers equal-area conic to Lambert azimuthal equal-area, while Europe and the conterminous United States and the tropics keep theirs. A criterion that changed every answer would be suspect and one that changed none would be decoration.
Fig. 1 Four regions, each with the projections ranked by the first-order criterion on the left of the pair and by the second-order one on the right. The rank correlations are 0.83, 0.81, 0.90 and 0.88 — closer than the whole-sphere figure’s 0.69 — and the winner changes in one region of the four: over a 30° cap the choice moves from the Albers conic to the Lambert azimuthal.

What a regional ranking finds that a global one does not

Three things, and the first is the least expected.

The two criteria agree more over a region than over the sphere. Correlations of 0.81 to 0.90 against the whole sphere’s 0.69. The reason is that a regional integral is dominated by a smaller range of latitudes, so the projections’ differences in pattern — where each puts its worst distortion — matter less, and their differences in overall severity matter more. Both criteria see severity the same way.

The winner nevertheless changes. Over a 30° cap the first-order criterion prefers the Albers equal-area conic and the second-order one the Lambert azimuthal equal-area, which is a real disagreement about a real choice: both are equal-area projections suited to a compact region, and the azimuthal’s advantage is entirely in how its geodesics bend.

Where they agree, they agree for different reasons. Over Europe both criteria put the Albers conic first, but the second-order ranking’s second and third places are not the first-order ranking’s, so a cartographer choosing a runner-up — because the winner is unavailable, or ugly, or already used by a neighbouring sheet — is choosing from a different shortlist.

The projection that has no second derivative

The first thing the regional measurement found was not about regions. It was about Robinson, and it invalidates a number in the ladder’s own earlier rung.

A projection with no second derivative. The flexion of Robinson and of Mollweide sampled every 0.5° of latitude. Mollweide has a formula behind it and its curve is smooth. Robinson is defined by twenty tabulated values with straight lines between them, so its meridian is a polygon: the flexion spikes at each multiple of 5°, where the table has an entry, to between 11 and 22 times its immediate neighbours' — 12.5 at the sharpest against 0.30 at the quietest. Neither number is a property of the projection Robinson designed; both are properties of the interpolation. Any second-order score for it reports where the sampling grid happened to fall, which is why it is absent from the rankings in this ladder and should be read that way in the earlier one.
Fig. 2 The flexion of Robinson and of Mollweide sampled every half-degree of latitude. Mollweide has a formula behind it and its curve is smooth. Robinson is defined by twenty tabulated values with straight lines between them, so its meridian is a polygon: the flexion spikes at every multiple of 5°, where the table has an entry, to between eleven and twenty-two times its immediate neighbours’.

The Robinson projection has no formula. It is a table of twenty values with an interpolation rule, and this site’s implementation interpolates linearly — a choice its own code documents, on the grounds that a cubic would be closer to Robinson’s intent and that linear keeps the derivative finite.

Linear interpolation keeps the first derivative finite and piecewise constant. It makes the second derivative a sum of spikes at the table’s own latitudes and zero between them. So a flexion measured at 25° is enormous and a flexion measured at 27.5° is nearly nothing, and neither is a property of the projection Robinson designed.

That means any second-order score for Robinson reports where the sampling grid happened to fall. The earlier rung’s whole-sphere ranking included it, and its position there should be read as this artefact — which is a correction to this site’s own published measurement, made by the machinery that produced it.

What the whole-sphere ranking said, for comparison

Eight world maps ranked by their second derivative. Each projection's root-mean-square flexion and skewness over the whole sphere, combined and ranked. The first-order ranking of the same eight by Kavrayskiy's criterion is given beside each bar; Spearman's rank correlation between the two orderings is 0.69. The two criteria are measuring different derivatives of the same maps and there is no reason for them to agree.
Fig. 3 The ladder’s third rung: eight projections ranked by their combined flexion and skewness over the whole sphere, with the first-order ranking of the same eight beside each bar and a Spearman correlation of 0.69 between them. The regional correlations measured above are 0.81 to 0.90, so restricting the question to a region brings the two criteria closer together rather than further apart — which is the opposite of what a reader would guess from the fact that the regional winners differ.

The reason the guess fails is worth one sentence: over the whole sphere a projection’s pattern of distortion dominates, and the two criteria weight patterns differently; over a region the pattern is nearly the same for every candidate, so what remains is severity, which they weight alike.

What the ladder can offer a reader in units they have

Every quantity in this ladder so far is a rate per radian of arc: exact, invariant, and impossible to hold. The regional measurement makes a translation available.

The bow of a drawn line is flexion × L² / 8 in map units, so at a stated map size the arc at which a ruled line departs from the true great circle by one pixel follows directly.

How long a ruled line stays within a pixel of the true route. On a map 1000 pixels wide showing the whole world, the arc a straight line drawn between two points can span before it departs from the great circle by more than 1 pixel, taken at the worst point of Europe. The relation is the ladder's own — the bow of a drawn line is the flexion times the square of the arc, over eight — so this table is the second derivative expressed as a distance. Gnomonic allows any arc at all, because its flexion is zero; Equirectangular allows 560 km. Every route drawn as a straight segment longer than its projection's number is a picture of a route nobody would fly.
Fig. 4 On a world map a thousand pixels wide, how long a straight line drawn between two points can be before it leaves the great circle by more than one pixel, taken at the worst point of Europe. The gnomonic allows any arc at all, because its flexion is zero — straight great circles is what it is for. The others allow between 560 and 1,400 kilometres, and every route drawn as a straight segment longer than its projection’s number is a picture of a route nobody would fly.

Five hundred and sixty kilometres on an equirectangular map is the sort of number a reader can act on. It is also the reason the site’s own figures draw great circles as computed curves rather than as line segments — a rule this ladder’s first rung established and this figure gives the numerical justification for.

The relation is checked rather than trusted: at the arc the flexion says is the limit, the drawn bow is measured at 0.90 pixels against the predicted 1, and at twice that arc it is four times as far, which is what a second-order law means.

The bow of a drawn line on Mollweide, against its length. A great circle drawn on Mollweide departs from the straight line between the same two points. The departure itself grows as the square of the arc — fitted exponent 2.00 — and the departure divided by the chord grows linearly, fitted at 1.00. The flexion recovered from the linear fit is 0.8617 against 0.8707 obtained by differencing the projection three times, which is a second route to the same number.
Fig. 5 The check, from the ladder’s first rung: a great circle’s departure from the ruled line between its ends, against the arc, on logarithmic axes. The departure grows as the square of the arc — fitted at 2.00 — and the flexion recovered from that fit is 0.8617 against 0.8707 obtained by differencing the projection three times. Two routes to the same number, which is what licenses converting a rate into a distance.

What the disagreement is made of

The two criteria disagree about the 30° cap, and it is worth taking that one case apart rather than leaving it as a table entry.

Both candidates are equal-area, so neither is paying anything in areal distortion. Their first-order difference is in shape: over a compact region the Albers conic’s two standard parallels let it hold the scale close to true over a band, while the Lambert azimuthal holds it true only at the centre — which is why the first-order criterion prefers the conic.

Their second-order difference runs the other way, and the reason is symmetry. The azimuthal projection is symmetric about the cap’s own centre, so its geodesics through the centre are straight and their bending grows outwards evenly; the conic’s symmetry is about an axis rather than a point, so a geodesic running across the cap bends differently depending on which way it runs. Aggregated over directions and over the region, the azimuthal comes out ahead.

That is a genuine cartographic statement rather than an artefact of the aggregation: for a compact region, the projection matched to the region’s shape wins the second-order comparison, and the projection matched to its extent in latitude wins the first-order one.

What was computed, and how

The two rankings use the same region and the same sample. The first-order criterion is Kavrayskiy’s, as everywhere on this site; the second-order one is the root-mean-square of flexion and skewness over directions, aggregated over the region with the area element, which is the same aggregation Airy’s and Kavrayskiy’s criteria use.

Robinson is excluded from every second-order ranking here, and the exclusion is enforced by the machinery rather than by editorial care: the assertion that finds its knots is part of the gate.

The ruled-line limit is computed at the worst point of the region, not the mean, because a map is used everywhere it covers.

The rank correlation is Spearman’s, computed over the projections both rankings could score — a projection that cannot show the whole region is dropped from both rather than scored on the part it likes.

How the site found it, which is the part worth copying

The Robinson result was not looked for. It appeared as an outlier in a completely different measurement: the next rung fits polynomial models to projections and compares the residual against the second-order magnitude, and four projections gave a ratio between 0.19 and 0.31 while Robinson gave 0.017 — a factor of twelve away from the others.

An outlier at a factor of twelve in a quantity expected to be constant is either a finding or a bug, and the way to tell is to measure the input rather than the output. Sampling Robinson’s flexion at half-degree steps took one line and settled it immediately.

That sequence — a ratio expected to be constant, an outlier, and a measurement of the input — is the site’s most productive habit and it is worth naming. A quantity that should be the same for every member of a family is a detector for anything wrong with any member, and it costs nothing to compute once the family is in place.

Two ways a criterion can be wrong, and one of them is invisible

The Robinson finding is worth generalising, because it is a species of failure this site keeps meeting.

A criterion can be wrong about a projection, which is visible: the number comes out too large or too small and a second route disagrees with it. This site catches those by computing everything twice.

A criterion can also be undefined for a projection while returning a number anyway, which is invisible: the machinery evaluates, the number is finite, the ranking sorts, and nothing anywhere reports that the quantity being ranked does not exist. That is what happened to Robinson’s second-order score, and it happened while the site’s own gate was running 267 assertions.

The general defence is the one this site already uses for its own claims — feed the machinery something it must refuse — and the specific defence added here is a check that a projection’s second derivative is continuous before its second-order score is quoted.

Flexion over Mollweide. The root-mean-square flexion over all directions, sampled at 375 points and drawn as a disc whose size is the value. It reaches 10.19 per radian of arc near 78°S 173°W. No first-order quantity — h, k, a, b, the areal factor or ω — carries any of this information.
Fig. 6 What a well-behaved second-order field looks like: the root-mean-square flexion of Mollweide over the sphere, each disc sized by its own value, reaching 10.19 per radian of arc in the far south-west. It varies smoothly from place to place, and it carries no information available from any first-order quantity — h, k, a, b, the areal factor or ω. A regional criterion is this field integrated over a region, and it is only meaningful where the field is a field.

Why the regional numbers are close, and what that means for a decision

A rank correlation of 0.85 is high enough that a cartographer choosing by the first-order criterion will usually be choosing well by the second. That is worth saying plainly, because the ladder could be read as arguing that everybody has been choosing projections wrongly, and the measurement does not support that.

What it supports is narrower and more useful. The two criteria are close on severity and differ on kind: a map that scores badly on the second order is one whose geodesics bend, which matters for a route map and does not matter for a choropleth. So the second-order criterion is not a better ranking; it is a ranking for a different purpose, which is the argument purpose before property makes in general and this ladder makes for one specific pair of derivatives.

The place it makes a difference is the compact region, where the two criteria’s winners diverge — and that is exactly where a cartographer has the most freedom, because a small region can be shown well by many projections and the choice among them is decided by small margins.

The same six projections, ranked over Europe and the tropics. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes third over the tropics. A table of projections ordered by distortion is a table about somebody's region.
Fig. 7 The first-order version of the same argument, from the choosing field: the same projections ranked over two regions, with the lines crossing between the columns. Adding a second criterion adds a third column to this picture for every region — and the correlations measured here say those third columns are close to the second ones, but not the same.

Where the model stops

Four regions. Europe, the conterminous United States, the tropics and a 30° cap, which are the site’s own standing regions. A wider survey would give a distribution of correlations rather than four numbers, and would probably find the winner changing in more places than one in four — the cap is the smallest region here and small regions are where the criteria diverge.

One weighting. Flexion and skewness are combined as a root mean square with equal weight, which the ladder’s third rung already shows to be a choice: the two halves of a second-order ranking disagree with each other more than the whole disagrees with the first-order one, at Spearman 0.45.

No aspect. Every ranking here is in the normal aspect. Since a rotation is an isometry both criteria are invariant under it at corresponding points, so an aspect search would move both scores together — but not by the same amount, since the regions are fixed and the maps move.

Nothing here re-measures the earlier ranking. The whole-sphere ranking in the ladder’s third rung is left as published, with this essay’s correction recorded rather than applied retrospectively — its other seven projections are unaffected, and Robinson’s position in it should simply be disregarded.

The ruled-line limit assumes one pixel and one map width. Both are parameters; the numbers scale as the square root of the tolerance and inversely as the square root of the map width, so a limit for a different map is one multiplication away and is not tabulated.

Flexion and skewness against direction on Mercator. At 20°E 40°N on Mercator, the rate at which the image of a geodesic turns (solid) and the rate at which the scale changes along it (light), both per radian of arc, drawn against the direction of travel about a zero circle. The largest flexion is 0.839 and the largest skewness 0.839, 90° apart. Neither is visible to Tissot's indicatrix, which describes only the first derivative.
Fig. 8 The object all of this integrates: flexion and skewness at one point on Mercator, drawn as polar curves against the direction of travel. A first-order quantity at a point is an ellipse and is fully described by two numbers; this is not an ellipse, and a regional criterion is a weighted average of several hundred of these — which is why the aggregation had to be defined before the ranking could mean anything.

Who found it, and when

Regional criteria are Airy’s, from 1861, and Kavrayskiy’s, from 1934, and both are integrals of a first-order quantity over a region. Nobody has published the second-order analogue, as far as this site’s own reading goes, and the reason is probably that the second-order quantities themselves are recent as a mapping tool — the flexion-and-skewness pair is standard in visualisation research rather than in cartography, where it appears in Goldberg and Gott’s 2007 work on map distortion measures.

Robinson’s projection is from 1963 and its table is the whole of its definition. The interpolation question has a documented history: the Robinson projection as implemented in different software packages differs, and the disagreements come from exactly this — some interpolate linearly, some with splines, and the results differ by amounts that are invisible on a printed map and enormous in a second derivative.

A projection defined by a table has no second derivative

The Robinson discrepancy between software packages is more than a curiosity about implementations, and it deserves stating as a general point about what a definition has to contain.

The projection’s definition is a table of values. Robinson gave a list of coordinates at intervals of latitude and nothing else, which is a complete specification of the projection at those latitudes and says nothing whatever about the points between them.

Interpolation supplies the rest, and it is not part of the definition. Linear interpolation, a cubic spline, a fitted polynomial, Aitken’s method — each reproduces the tabulated values exactly, each gives a different map between them, and every one of them is a legitimate reading of the same specification.

The values agree and the derivatives do not. Between two tabulated latitudes, linear and spline interpolation differ by an amount invisible on a printed sheet — the maps look identical, and for every first-order purpose they are. Differentiate twice and the difference is not small: a piecewise-linear reading has zero second derivative inside each interval and a delta at every node, while a spline has a continuous one. The two are not close; they are different kinds of object.

So a table-defined projection has a well-defined position, an approximately defined first derivative, and no second derivative at all until an interpolation rule is named. That is not a defect in Robinson’s work — he was specifying a map to be drawn, and for drawing a map the table is enough.

It becomes a defect the moment the projection is measured rather than drawn, which is what this rung does. A flexion figure for Robinson is a figure for Robinson-plus-an-interpolation, and two papers quoting different figures may both be right about different objects with the same name.

And the same question should be asked of every projection defined by data rather than by a formula. Robinson is the famous case and it is not the only one: any projection specified as a table, a set of control points, or a fitted set of coefficients with a stated evaluation scheme carries its interpolation inside its definition, and whether that interpolation is smooth enough to differentiate twice is a separate question from whether it reproduces the data.

Which sharpens what a measurement of such a projection can claim. A first-order quantity — a scale factor, an angular deformation — depends on the interpolation only weakly, because any reasonable rule matches the tabulated values and has nearly the right slope between them. A second-order quantity depends on it entirely. So the same table supports a distortion comparison and does not support a flexion comparison, and nothing about the table announces where the line is.

That is a general statement about specifications rather than about this projection. A definition adequate for the use it was written for can be inadequate for a use invented later, and the inadequacy is not visible in the specification: Robinson’s table is exactly as complete as it ever was. What changed is that somebody started differentiating it twice.

The remedy is one line in the specification: a table-defined projection should name its interpolation rule, and a measurement of one should name the rule it used. Neither happens, and the second is cheaper and entirely within the measurer’s control.

Where the ladder goes next

This rung has the second derivative aggregated, compared and translated into a distance. What it has not done is connect it to the thing a practitioner actually fits.

Every georeferencing tool in existence fits a polynomial between two coordinate systems — affine, quadratic, sometimes cubic — and the choice is usually made by counting control points. What that choice buys is an order of convergence, and the first term an affine model cannot hold is precisely the second derivative this ladder measures.

Where this goes next

Eight world maps ranked by their second derivative. Each projection's root-mean-square flexion and skewness over the whole sphere, combined and ranked. The first-order ranking of the same eight by Kavrayskiy's criterion is given beside each bar; Spearman's rank correlation between the two orderings is 0.69. The two criteria are measuring different derivatives of the same maps and there is no reason for them to agree. The second derivative has its own ranking Rank eight world maps by how much they stretch and Mercator comes sixth of eight. Rank the same eight by how much they bend and it comes third. The two halves of the second-order score disagree with each other more sharply than either disagrees with the first-order one — Spearman 0.45 against 0.69. The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region. Distortion over a region An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion. A great circle and the ruled line, on Mercator. The shorter of the two routes is the curved one. The great circle between the two marked points is drawn against the straight line a ruler would give between them on Mercator; over 70° of arc the curve departs from the ruled line by 11.9 per cent of the chord. The flexion at the midpoint, which is the rate the image turns per radian of arc, is 0.788; the angular deformation there is 0.00°. Tissot stops at the first derivative Every quantity this site has measured is read off one derivative of the projection. A map can be conformal at a point — the indicatrix a circle, the angular deformation zero to eleven figures — and still bend every geodesic through it, at a rate of 0.839 radians of turning per radian of arc.

Named alongside this one

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CorrelationDistortion measuresFlexionInterpolationKavrayskiy's criterionRankingRegional distortionRobinsonRoute planningSecond-orderSkewnessTolerance