Gradient — the ladder · Cartographic Projection

Tissot's ellipse and the one that governs gradients, at 20°E 48°N. The solid ellipse is the image of a small circle — Tissot's indicatrix, semi-axes a and b. The dashed one is the image of a unit gradient, whose semi-axes are 1/b and 1/a because a gradient transforms by the inverse transpose of the Jacobian rather than by the Jacobian. Its long axis therefore lies where the indicatrix's short one does. On Mercator both are circles, so a gradient's direction survives; on Lambert cylindrical the two ellipses are the same shape turned through a right angle, so the worst direction for a gradient is the best direction for a shape.

A slope is not a shape

Every map in this collection has carried geometry. An applied map far more often carries a field — elevation, pressure, a density — and the first thing anybody does with one is differentiate it. A gradient is a covector, it transforms by the inverse transpose of the Jacobian, and the ellipse that governs it is the indicatrix turned inside out.

rung 1 · distortion

The same six contours on Mercator and Lambert cylindrical. The value of a harmonic sum with a summit and a basin in the northern mid-latitudes travels with the point, so the set of points at a stated level is the same set on every map and each contour is exactly right on both panels. Everything a reader measures from them is not: the spacing between neighbouring contours, their lengths, and the area between two of them all change from one panel to the other, and the two panels are the same field.

The contour is right and the reading is wrong

There is exactly one thing about a field that no projection can get wrong: which points share a value. The contour lines on any two maps of the same field are the same set of points. Every quantity a reader takes off them — the spacing, the length, the area between two of them, the hypsometric curve — is not, and the two kinds of map get different ones wrong.

rung 2 · distortion

Steepest descent on Lambert cylindrical, computed on the ground and on the page. Fifteen routes, each started at the same place twice. The solid line follows the true direction of steepest descent on the sphere; the dashed line follows the direction read off the page at every step, which is what an analysis of a projected grid does. Both take the same length of step on the ground, so the only difference between them is the direction. They part by up to 1517.8 kilometres, against a bearing error of 45.6°. The faint lines are contours of the field, which is seven caps at stated centres and widths.

Water runs downhill on the ground, not on the page

A drainage network is the set of steepest-descent trajectories of a field, so it is built entirely out of directions. A conformal map preserves those directions exactly and therefore preserves the whole network; an equal-area map does not, and sends a route up to 892 kilometres away from where the water actually goes.

rung 3 · distortion

A finer grid makes a measured slope worse. The error in the direction of steepest ascent, against the spacing the field was sampled at, for four noise levels. With exact values the curve falls at a fitted slope of 2.00 — second order, which is what a central difference is. Add noise and the same curve turns over: a finite difference divides the noise by the spacing, so halving the grid doubles the noise in the slope while quartering an error that was already negligible. The minimum is where the two meet, and it is not at the fine end.

The slope of a field that was measured

Three rungs differentiate a formula, which is what makes the projection the only thing under test. A real field is a grid of numbers with an error on each of them, and differencing such a thing divides the noise by the spacing — so a finer grid gives a worse slope, there is a best spacing, and it is the cube root of the noise.

rung 4 · distortion

A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere.

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

rung 5 · distortion

Conformality helps and does not save it. The proportion of a stated terrain whose plan curvature changes sign when it is read off a grid in each projection. A conformal map turns every direction through the same angle, so the contour and the slope line stay perpendicular and the sign ought to survive — and it mostly does, at 0.30 per cent against 13.3. It is not zero, and the term that flips it is the gradient of the scale factor: the curvature of a curve under a conformal map is (κ − ∂ₙ log λ)/λ, and Mercator's λ has a gradient.

The curvature of a field is not the curvature of its picture

Whether a place is a spur or a hollow is the sign of a second derivative, and every automatic terrain classification is built on it. Read off an equal-area grid, that sign is wrong on 13.25 per cent of a sheet — and conformality reduces it to 0.30 per cent without removing it.

rung 6 · distortion

The points a projection cannot move. The critical points of a stated field, found twice: once on the sphere from its own gradient, and once in Lambert cylindrical's page coordinates from the page's own numbers, with nothing shared between the two searches. seven points, 3 maxima, 2 saddles and 2 minima, and the two sets agree in position to 2.3e-7 degrees and in type at every one. The page's Jacobian is invertible wherever the map is a map, so it sends a zero gradient to a zero gradient and cannot move a critical point anywhere.

What the page cannot move

Six rungs measure what a page does to a field's readings and every one of them moves. The critical points do not: found on the sphere and found again in a projection's own page coordinates with nothing shared between the searches, they agree to 10⁻⁷ degrees and in type at every point, on every projection. What the page does move is their shape, by a factor bounded exactly by the indicatrix's axis ratio squared.

rung 7 · distortion

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.

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.

rung 8 · distortion