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

Conservation — where it appears

The property that an operation preserves a total, which is what area-weighted rebinning between two cell schemes does and very nearly all it does. It is exact when the overlap areas have closed forms and only as good as the arithmetic when they have to be clipped.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

A triangle of 25.0° excess, drawn on Mercator. The three geodesic sides as curves and the three straight sides a ruler draws, with each vertex labelled by how far the ruler's angle is from the true one. The straight-sided triangle's angles sum to exactly 180° because it lies in a plane, and the real one's sum to 205.02°, so the three errors have to account for the whole 25.02° of spherical excess between them — and they do so on this conformal projection exactly as they do on any other.

Conformal does not mean the angles are right

A conformal projection preserves angles between curves at a point. Draw a triangle on one with a ruler and its angles are wrong by degrees — and the total error is fixed before the projection is chosen, because a plane triangle sums to 180° and the real one does not.

wrong · Audit Normal gravity, derived from four constants. Gravity on the surface of the level ellipsoid, by Somigliana's closed form, for WGS84. Nothing here is measured: a, f, GM and ω go in and the whole curve comes out, rising 5186 milligal — 0.53 per cent — from equator to pole. The two open marks are the published values of equatorial and polar gravity for WGS84, which the derivation reproduces to ten significant figures rather than borrowing.

The ellipsoid is a level surface

WGS84 publishes two dozen constants and defines four of them. The other twenty are consequences — polar gravity, the potential of the ellipsoid, the coefficient that dominates the Earth's gravity field — and every one comes back here from a, f, GM and ω to the last digit published.

datums · Height Why a levelled height is not a distance. The correction between raw levelling and orthometric height, for lines at 200, 500, 1000, 2000 metres above the geoid running north from 50°. It is not instrument error: level surfaces converge towards the pole, so a run that stays on one of them gains height relative to another. A line 2000 metres up reaches 647 millimetres over 400 kilometres. The dashed line is 10 millimetres, which is about what a first-order levelling network closes to over that distance — so this is not a refinement, it is the larger of the two numbers.

A levelled height is not a distance

Level surfaces converge towards the poles by five metres in a thousand, so a chain of perfectly executed levelling observations does not sum to a height difference. The correction over four hundred kilometres of northing is larger than the network's own closure.

datums · Height Two readings of the same triangle, and the band between them. The same three vertices, joined two ways: with straight lines in the plane the coordinates are stored in, and along the ground. Every dot is a point the two readings disagree about — inside on one and outside on the other. The band covers 29.5 per cent of the polygon, which is 4241 thousand square kilometres, and it is not an error in either reading: the file does not say which one it means. Densifying the stored boundary removes it, which is the only fix there is. Drawn in Equirectangular.

Inside is a claim about the edges

Whether a point is inside a polygon is not a property of the point and the polygon. It is a property of the plane the edges were understood to be straight in, and between two readings of the same file there is a band of disagreement — 29 per cent of one triangle's area, 4.2 million square kilometres, and the file does not say which reading it means.

applied · Dataset One field, one round trip between two cell schemes. Left: a stated field binned into an equal-angle grid of 36 by 18 cells. Right: the same field after being rebinned into an equal-area grid of 30 by 15 offset by six degrees of longitude, and rebinned back. Every step is exact area-weighted averaging, the total is preserved to 2 × 10⁻¹⁶, and the root-mean-square difference between the two pictures is 0.144 on a field whose own standard deviation is 0.370.

The same data on two grids

Five essays have addressed, queried and ordered cells within one scheme and nobody has moved a number between two. Doing it exactly — area-weighted, both directions — preserves the total to 2 × 10⁻¹⁶ and loses 39 per cent of the field's own standard deviation in a single round trip; six round trips leave 23 per cent of its variance. The quantity that would reveal the damage is the one that never moves.

applied · Cells The piece two schemes share, clipped rather than assumed. A cell of a gnomonic cube, whose four edges are great-circle arcs because a straight line on a gnomonic face is one, against a cell of a longitude–latitude grid, whose north and south edges are parallels and are not. Their overlap is neither a rectangle nor a spherical polygon of any standard kind, and it is 5965687 km² of the cube cell's 5965687 km² — 100.0 per cent. Computing it needs the arc of one boundary intersected with the plane of the other, which is three equations and two roots, and it is exact.

Cells that are rectangles in no coordinate

The previous rung measured what moving a field between two cell schemes costs, and did it between two schemes whose cells are longitude–latitude rectangles — which is what made every overlap a rectangle with a closed-form area. The schemes anybody actually argues about have cells that are rectangles in no coordinate, and their overlaps have to be clipped.

applied · Cells Two source geometries of 96 cells each, rebinned to the same three targets. Both curves start from a source of 96 cells and rebin to targets of 32, 128, 512 cells, so the count ratio is identical along them and the only difference is the shape of the source cells: gnomonic squares on a cube against rectangles in longitude and latitude. The ratio dominates — both curves fall by more than half across the range — and the shapes still separate by 25 points at the middle target. The cube loses less, because its cells are all much the same size and the lon/lat source's collapse towards the poles.

The same number of cells, in two shapes

Moving a field between two cell schemes loses 18 per cent of it per cell in one geometry and 39 in another, and the earlier measurement could not say whether that was the shape of the cells or the ratio of their sizes, because changing the schemes changed both. Holding the counts settles it: the count ratio decides most of the loss, and the shape is still worth a quarter of the field.

applied · Cells What a rebinning loses depends on where the target's edges are. Two grids of fixed counts, fixed shapes and fixed resolution, with the target slid across the source from perfect alignment to a full cell. Nothing about either grid changes except where its boundaries fall. The loss runs from 27.5 per cent at zero to 56.3 at half a cell — a factor of 2.04 — and the longitude-only curve returns to its starting value at a full cell to six decimal places, which is the periodicity check. A cell boundary that coincides with a target boundary loses nothing, and a grid comparison that does not say where its boundaries are has left that out.

When the edges do not line up

Rung eight held the cell counts equal so that shape could be compared without the count ratio drowning it, and recorded a doubt: a longitude–latitude source shares its boundaries with a longitude–latitude target wherever their counts share a factor. The mechanism is real and worth a factor of two. It was not what the published number was made of.

applied · Cells 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.

distortion · Gradient

Named alongside it

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

VerificationAggregationCell systemDiscrete global gridInterpolationRebinningAreaCellClippingConformalityEqual-areaEquipotential

All concepts