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The thread: What survives a change of coordinates — page 5

Scale along the meridian and scale along the parallel depend on how the sphere was parameterised. The principal scales, the areal factor and the angular deformation do not, and only those are properties of the map itself. Essays 97 to 100 of 100.

Densifying helps one and hurts the other. The standard deviation of the shoelace area, and the bias of the perimeter, against the number of vertices on the same 100-metre circle with the same 5-centimetre noise on each. The area's spread falls from 9.9 m² to 2.0 — the closed form says it goes as the square root of the vertex count's reciprocal, because a vertex's influence on the area is the vector between its two neighbours and densifying shortens it. The perimeter's bias rises from 0.07 mm to 261, a factor of 3697, because every leg contributes its own σ²/d and shorter legs contribute more.

Measuring distortion

The area is unbiased and the perimeter is not

A boundary measured from noisy vertices comes out long, always, by σ²/d on every leg. The area enclosed by the same vertices comes out exactly right, because a shoelace is bilinear and the cross terms vanish. So densifying a boundary makes its area five times more precise and its perimeter three thousand times more wrong, and every compactness score computed from it falls short.

Two maps with the same ellipses. Mollweide, and the same projection with the sheet turned 37° on the desk. Every ellipse is the same size and the same shape as its counterpart, and its angle to its own graticule is the same — the two indicatrix fields agree to 1.8e-12, which is arithmetic noise. The rotation differs by 37° at every point. So the classical description of a map's distortion is exactly blind to the difference between these two, and the difference is the whole of what a surveyor calls convergence.

Measuring distortion

The fourth number the ellipse does not carry

A Jacobian has four independent entries and an indicatrix reports three. The missing one is the rigid rotation in the polar decomposition A = R·S, the indicatrix is exactly S, and R is what turns north into grid north: for the transverse Mercator it agrees with the survey formula for convergence to 5 × 10⁻¹⁰ degrees, from a different library and a different derivative.

What a reprojection does, drawn where it does it. The angular deformation of the map that takes Sinusoidal's page to Gall–Peters's — which is what a reprojection is — shaded over the sphere and drawn on Mollweide so that equal ground areas are equal page areas. Both parents are equal-area, so the composite preserves area exactly; its mean angular deformation is 55.5° against 38.9° and 32.2° for the two maps separately. Composing two maps did not average their shape errors; it made a larger one.

Measuring distortion

Two indicatrices do not make a third

Reprojecting is composing, and the composite's indicatrix is not a function of its parents'. Mollweide followed by Hammer is gentler than either map over eighty per cent of the sphere; the sinusoidal followed by Gall–Peters is worse than both everywhere. What separates them is one angle, and it is the number rung ten showed the ellipse does not carry.

One current, two maps, and only one of them conserves it. A flow that has no sources or sinks anywhere on the sphere — it is the perpendicular of a stream function's gradient, so its divergence is zero by construction — drawn on two projections. The arrows on the right are the images of the same ground velocities as the ones on the left. On the equal-area map the drawn field is still divergence-free; on the other it is not, and the arrows drawn in the warning colour are where a reader measuring the picture would find a source or a sink that is not there.

Measuring distortion

A current drawn on a page has sources

Seven rungs project a scalar field and ask what the page does to its gradient. A wind or a current is the other half of what gets mapped, and the operator that matters for it is the divergence — which is preserved by an equal-area map exactly, by no other map at all, and by a conformal map least of anybody's expectation.

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