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

Fitting a polynomial to Mollweide, and what each order buys

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

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.

It is one of 4 figures drawn by the same construction, and the drawing above is this one with nothing changed.

3 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.

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. Measuring distortion

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.

 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.

 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.

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