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

Skewness — where it appears

How fast the local scale changes along a geodesic, per radian of arc; the companion of flexion and the other half of a map's second-order failure. Together with flexion it is the whole of a projection's second-order behaviour, and neither of them appears in any classical distortion measure.

Named by 8 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 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 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

Named alongside it

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

Second-orderFlexionToleranceJacobianNumerical differentiationRankingTissot's indicatrixConformalityConvergence orderCorrelationGeodesicIndicatrix

All concepts