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

Second-order — where it appears

The part of a projection's behaviour carried by its second derivative: the bending of curves and the rate the scale changes along them. It is the pair of quantities Tissot's indicatrix cannot express: flexion, which is how fast an image curve turns, and skewness, which is how fast its scale changes.

Named by 11 essays across one field — each of them below, with the objects they name alongside it.

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.

distortion · Flexion Six projections, at 40° north. Flexion (solid) and skewness (light) against direction of travel, drawn about a zero circle at one point of each projection. The three conformal ones have curves of exactly equal size, offset by exactly 90°, because on a conformal map both quantities are components of one vector — the gradient of the log of the scale. The others have neither property: ratios run from 0.72 to 2.72.

Bending and stretching are one failure

The ladder was written expecting flexion and skewness to be two independent ways for a map to be wrong. On a conformal projection they are not independent at all: the two extremes are the same size to four figures and sit exactly ninety degrees apart, because both are components of a single vector.

distortion · Flexion 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.

distortion · Flexion 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.

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.

distortion · Flexion Fitting a polynomial to Mollweide, and what each order buys. An affine, a quadratic and a cubic transformation fitted by least squares between the sphere and Mollweide over patches from 8° down to 0.5° radius, centred at 20°E 40°N. Each is a straight line on these axes and its slope is one more than its own degree: 1 → 2.00, 2 → 3.00, 3 → 4.00. That is not a coincidence and it is this ladder's subject: the first thing a model of degree d cannot represent is the term of degree d+1, so the affine model's error is governed by the second derivative — the flexion and skewness measured everywhere else here.

A local model has an order

Every georeferencing tool fits a polynomial between two coordinate systems and the choice of degree is usually made by counting control points. What it buys is an order of convergence — 2, 3 and 4, measured — and the first term an affine model cannot hold is the second derivative this ladder has spent five essays on.

distortion · Flexion The same map on four differently prepared pages. Mollweide at 30°E 20°N, then the same projection with a rotation and a magnification applied to the page, then with one axis stretched by 1.6, then with a shear of 0.5. The similarity changes nothing: flexion, skewness, ω and the anisotropy are identical to every printed figure, and only the last column — the same turning measured per unit of page arc rather than per unit of ground arc — moves, by exactly the magnification. The stretch and the shear change all of them, and flexion by 35 per cent.

The second derivative is not an invariant

The first-order ladder established which quantities survive a change of coordinates and which are artefacts of the parameterisation. Asked of the second order, the answer is that flexion survives a rotation and a magnification of the page exactly, and survives nothing else: a stretch of 1.6 in one axis moves it by eleven per cent, a shear of 0.5 by thirty-five, and a shear of 3 leaves a ranking of eight world maps with a rank correlation of −0.07 to the one it started with.

distortion · Flexion 10,000 observations of one place, averaged 120 times. Left: 400 single observations of the same place, with a 60-kilometre standard deviation east and north, drawn across ±400 km. They are scattered about the truth and their average is unbiased on the ground. Right: 120 independent averages of 10,000 such observations each, computed on the page and taken back to the ground, drawn across ±1.8 km. The cloud is tight, as averaging ten thousand things should make it, and it is not centred on the cross: it sits 489 m away, against 403 m predicted by the projection's second derivative alone.

The average of noisy positions moves

Average sixty thousand scattered observations of one place on a Mercator map and the answer is 404 metres too far north — at every sample size, because it is a bias and not noise. The same average on the Lambert cylindrical equal-area is 404 metres too far south, the two being ½ (σ²/R) tan φ and its exact negative, and on the plate carrée it is not displaced at all.

distortion · Precision A geodesic circle on Mollweide, and the two models of where it goes. A circle of geodesic radius ρ about 20°E 45°N, projected. The dashed outline is where the indicatrix says it goes — the ellipse a Tissot figure draws — and the thin solid one adds the quadratic term, which is the flexion and skewness this ladder measures. At 16° the indicatrix is out by 13.5 per cent of the figure's own size and the second-order model by 2.20 — a factor of 6.16, which is what one more derivative buys.

The size at which the second derivative arrives

Six essays have measured a projection's second derivative at points, over regions and under transformations, and none of them says at what size it stops being a curiosity. The answer needs a figure with an extent rather than a point, and it is smaller than anybody drawing a national map would guess: the indicatrix alone places a shape to one part in a thousand out to thirteen kilometres.

distortion · Flexion One degenerate zero, nudged, becomes two ordinary ones. The direction of steepest ascent within twelve degrees of the north pole, for the sectoral harmonic alone and with two amounts of the tesseral added. On the left is one zero of index −2, a monkey saddle: three ways up and three ways down, and a Hessian that vanishes. On the right are two ordinary saddles of index −1 each, both of which the second-derivative test names correctly. Nothing has been added to the field but a term whose size can be made as small as anyone likes, and the classification changes at every nonzero value of it while the total does not change at all.

The second derivative cannot classify

Seven essays have used a second derivative to measure a size. The one thing a second derivative is classically used to do is say what KIND of thing is at a point, and on a sphere that use fails: the determinant test totals six where the truth is two, its failure is confined to exactly the field it is asked about, and the count that gets it right never differentiates twice.

distortion · Flexion A 900 km circular accuracy at 55° north, projected. Six thousand ground positions drawn from a circular error of 900 kilometres about one place, each projected in Mercator and plotted as a displacement from the projected place. The curve is the nominal 95 per cent ellipse, computed the standard way — the ground covariance sandwiched between the projection's own derivatives. It holds 93.83 per cent of the points, the cloud is measurably longer than it along its own long axis by 5.52 per cent, and it is not symmetric: the third moment along the page's second axis is 0.727 rather than zero.

The error ellipse is not an ellipse

Rung two pushed a covariance through a projection with the same matrix sandwich that draws an indicatrix. That is a first-order operation on a map with a second derivative, so the propagated distribution is not the ellipse the sandwich draws — and a nominal 95 per cent ellipse holds 93.06 per cent on one projection and 95.63 on another, in opposite directions, from the same input.

distortion · Precision The bias is σ² over the length, over two decades. The amount by which a measured baseline is longer than the true one, against its length, for a twenty-millimetre error on each end. The line is the second-order prediction σ²/d. The measured bias times the length is constant to a factor of 1.0014 across the whole range, and it sits 8.2 per cent below the prediction — which is the fourth-order term the expansion drops. The estimate is antithetic, so the first-order scatter cancels exactly and a bias of thirty-seven microns is measured at a t-statistic of 349.

A length measured from noisy points is too long

A distance is a square root, a square root is concave, and the average of the distances is not the distance between the averages. The gap is a bias with one sign: 37 microns on a ten-metre baseline with twenty-millimetre marks, following σ² over the length across two decades, and it adds rather than cancelling — so the same boundary is 1.5 parts per million longer when it is measured in more pieces.

distortion · Precision

Named alongside it

The objects these essays reach for when they reach for this one.

FlexionSkewnessNumerical differentiationToleranceJacobianPrecisionRankingTaylor expansionTissot's indicatrixBiasConformalityConvergence order

All concepts