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

Flexion over Mollweide

Drawn here at the parameters it defaults to, with every essay that calls it.
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.

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.

It is drawn by secondorder-figure with show: "second-order-field" — one member of a family of 4 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

9 essays call it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

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°. Measuring distortion

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.

 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. Measuring distortion

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.

 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. Measuring distortion

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.

 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. 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.

 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. Measuring distortion

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.

 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. Measuring distortion

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.

 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. Measuring distortion

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.

 Two compromises along one path, and they do not agree. The angular deformation and the flexion of every blend between Mercator and Lambert cylindrical, each divided by the straight line between the two parents' own values. A value of one means the blend is exactly the average of the two errors and has bought nothing. The angular deformation dips to 0.649 at a weight of 0.13; the flexion dips only to 0.955, and it does so at 0.50. Anybody choosing a compromise is choosing a weight, and the two orders want different ones. Measuring distortion

The best compromise for angle is not the best for bending

Every compromise projection in the library is an average of two others, and averaging is a first-order operation — so the second derivative was never part of the bargain. Swept along five ordinary blend paths, the weight that minimises angular deformation and the weight that minimises flexion are between a quarter and a half of the axis apart, and how much a compromise buys at one order predicts nothing about the other.

 Two special lines, and they are not the same line. Each projection's true-scale parallels, found by solving for a parallel scale of exactly one, and the parallels along which its second-order failure vanishes, found by minimising the flexion. Five of eleven put them more than five degrees apart, and Gall–Peters puts them forty-five. The pattern is tangency: a projection with one standard parallel has both lines there, and a secant one has its true-scale lines moved off the centre while the bending zero stays. Measuring distortion

The lines where the bending vanishes

Every projection has a line printed in its margin — the standard parallel, where the scale is exactly one. It has a second special line nobody prints: the one along which a geodesic is drawn straight to second order. On Gall–Peters they are forty-five degrees apart, and the rule turns out to be tangency — a tangent construction puts both lines at its point of contact and a secant one moves only the first.

The whole library