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

Numerical differentiation — where it appears

Taking a derivative by differencing a function rather than by algebra, at a step size that sets the smallest effect the result can resolve. Its step size sets the smallest effect the result can resolve, so an assertion built on it has a noise floor that has to be stated with it.

Named by 8 essays across 2 fields — each of them below, with the objects they name alongside it.

Every projection in the library, measured against both properties. Maximum angular deformation across the bottom, maximum areal error up the side, both on logarithmic scales and both measured over several hundred sample points rather than taken from the projection's description. Conformal projections lie on the left edge, equal-area ones along the bottom, and the corner where both vanish is empty because a projection there would be an isometry of the sphere onto the plane. All 26 are plotted; the six named here are Mercator, Web Mercator, Gall–Peters, Mollweide, Equirectangular, Stereographic.

Measuring instead of naming

A projection is called conformal because that is its name. Running the definition as a computation over several hundred points takes about twenty lines, catches a projection the whole internet uses, and is almost never done.

distortion · Tissot The image of a circle on Mercator, against its own indicatrix. A circle of three radii on the ground at 30°, 40°, projected exactly — the solid curve — against the ellipse the indicatrix predicts for it, dashed. The filled dot is the image of the circle's centre and the hollow one is the centre of area of what was actually drawn, which is not the same point. The departure runs from 3.14 per cent at 4° to 15.80 per cent at 16°, so it grows in proportion to the radius rather than to its square: halving the circle halves the relative error and does not quarter it.

The indicatrix is a limit

Tissot's ellipse describes an infinitesimal circle, and every published one is drawn finite. The error of the description falls as the radius rather than as its square — halving the circle halves the lie — and on the Robinson projection at a table entry it does not fall at all: sixty metres and six kilometres are both 1.17 per cent wrong.

distortion · Tissot 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 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 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 Every projection's angular deformation, and the tolerance that judges it. The whole library on one logarithmic axis, with the tolerance drawn as a line. The population is bimodal: the projections that satisfy the condition sit at 2.09e-6 and below, the ones that do not at 3.85e-1 and above, and there is nothing between. The tolerance could be moved anywhere in that gap — a factor of 1.84e+5 — without changing one verdict. Larger marks are projections that claim the property.

The tolerance that decides the verdict

Eight rungs of this ladder hand out verdicts, and every one rests on a tolerance chosen once, in the site's second phase, at sixty times a measured noise floor. Swept, it decides nothing: the population is bimodal, the tolerance sits in a gap 238,000 times wide for conformality and 646 million times wide for equal area, and the verdict with the least room is a passing one whose margin is the arithmetic's rather than the map's.

wrong · Audit 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

Named alongside it

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

FlexionSecond-orderToleranceConformalityInfinitesimalJacobianQuadratic lawSkewnessTissot's indicatrixVerificationAssertionAudit

All concepts