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

Resolution — where it appears

The finest detail a representation can hold, which for a raster is its cell size and for a stored coordinate is the value of its last written digit. Below it a curve is a lattice walk, and any quantity measured there is a measurement of the lattice.

Named by 17 essays across 7 fields — each of them below, with the objects they name alongside it.

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.

distortion · Gradient One unit of a vector tile, in metres of ground. A vector tile's coordinates are integers on a lattice 4096 units across the tile, and the tile halves at every level, so one unit is a distance that halves too: 5.48 m at z10 and 0.086 m at z16, at 55°. It is also a different distance at every latitude, by cos φ, because the tile is in Web Mercator — the same factor that makes a grid metre a different quantity of ground at every latitude, arriving in the file format rather than in the projection.

A vector tile has an integer grid

Six essays on this ladder treat a vector tile as the thing a raster tile is not: geometry, resolution-free, styled at draw time. Its coordinates are integers on a lattice 4,096 units across a tile, the tile halves at every level, and at 55° north one unit is 88 metres at zoom 6 and 21 millimetres at zoom 18.

applied · Screen What a geoid model leaves out, against the degree it stops at. The RMS of everything above the model's highest degree, from Kaula's rule — the statement that the normalised coefficients at degree n are about 10⁻⁵/n². The line is R × 10⁻⁵ ÷ n, so a model to degree 360 omits 17.7 centimetres and one to 2190 omits 2.9. Every orthometric height derived from such a model carries that as an error, and it is not quoted with the height.

The geoid model stops at a degree

Eleven essays treat the geoid as a surface that exists. Every geoid anybody uses is a series truncated at a degree, so every orthometric height derived from one carries an omission error nobody quotes with the height — eighteen centimetres at degree 360 — and the same truncation removes two thirds of the slope, which does not converge at all.

datums · Height A curve built to have dimension 1.2619. The generator replaces every segment with four of equal length at headings 0, +60.0°, −60.0° and 0. Closing the displacement fixes the length ratio at 0.33335, and four copies at that ratio give a dimension of exactly log 4 / log(2 + 2 cos θ) = 1.2619. Nothing here is measured yet: this is the construction the measurement will be checked against. Drawn at depth 5, which is 1024 segments, with the second-level shape shown faint beneath it.

A line has a length only at a scale

A curve given by a formula can be sampled as finely as the picture needs. A map is not that: the geometry that reaches the page has been through an algorithm whose job is to throw most of it away. The first thing that goes is the idea that the line had a length.

applied · Generalise Where the summits are, and where the water divides. A stated range 100 km across and 80 km long, its summits 3,000 m high and about 35 km from the western edge, with a plateau falling gently to the eastern edge. Over 40 km of the range the plateau carries a broad rise up to 18 km east of the summits, and a gorge cuts through the range in the middle of that stretch. The dashed line runs through the highest summits; the solid line is the watershed, computed from the land. Lightly shaded ground drains west. Where they part, the darker ground lies east of the summits and drains west: 331 km², reaching 18.1 km from the summits. The terrain is stated, not surveyed.

The highest summits stay where they are, and the watershed does not

A treaty that puts a boundary on the highest crests which divide the waters names one line where the water divides on the summits and two where a river has cut back through the range. On a stated range the two part by up to eighteen kilometres. They also differ in kind: an error of ten metres in every height moves the summits three metres and the watershed on the plateau more than a kilometre.

practice · Boundary What simplifying a boundary does to the number stored beside it. One region, simplified at five tolerances, with the error in the two quantities a consumer computes from the pair. If the density was stored, the total it implies moves by exactly the area's error — -1.55 per cent at the loosest tolerance. If the total was stored, the density it implies moves the other way by the same amount. Nothing in the file says which of the two was measured and which is being derived, and the simplification is normally done by a tool that never opens the attribute table.

The attribute is a claim about the geometry

Fourteen essays price what a stored coordinate means and not one asks what the number stored beside it means. A rate is a quantity divided by an area, the area belongs to the geometry, and no format records which area — so a simplification that moves the outline by nothing visible moves the implied total by 1.55 per cent, an unweighted average of densities is 4.09 per cent out, and a choropleth gives a polar square kilometre fifteen times the ink of an equatorial one.

applied · Dataset 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 The geoid's curvature is whatever degree the model was stopped at. The Gaussian curvature a geoid model contains, as a fraction of the sphere's own, against the degree the spherical harmonic series is truncated at. It rises in proportion to the cutoff, because a degree-n undulation moves the curvature by n(n+1) − 2 times its own amplitude while the amplitude itself falls only as n to the three halves. The dashed line is the ellipsoid's whole variation from equator to pole, 1.33 per cent; the geoid overtakes it at degree 1332, a half-wavelength of 15 km. Below that resolution the ellipsoid's flattening is the larger term and above it the geoid's own bumps are.

The geoid has a curvature only if you say where you stopped

Thirteen measurements have priced the curvature of surfaces that are stated — a sphere, an ellipsoid, a triaxial body, a surface with a hole, real relief. The surface a height actually refers to is none of them. It is an equipotential of the Earth's own gravity, it is only ever given as a series stopped at a degree, and its Gaussian curvature carries two more powers of the degree than its height does. So the omitted height converges, the omitted slope diverges as a logarithm, and the omitted curvature diverges as a power: EGM96 carries 0.36 per cent of the sphere's own curvature, EGM2008 carries 2.19, and a one-kilometre model would carry twenty.

impossibility · Curvature 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 The same 2-pixel road at three latitudes, zoom 5. The dark bar is the mark as drawn — 2 pixels, identical in all three panels, because that is what the stylesheet says. The pale band behind it is the ground that mark covers, drawn to one common ground scale: 9.78 kilometres at the equator, 6.92 at 45° and 1.70 at 80°. The reader sees the dark bar and is being told about the pale one.

The road is drawn two pixels wide

Seven rungs measure what a screen map does to position. Nothing on a map is a point: every mark has a width, the width is chosen in pixels, and a two-pixel road covers 9.78 kilometres of ground at the equator and 1.70 at 80° north. That is a generalisation applied at a strength varying by a factor of six across one sheet, by a stylesheet with no latitude in it.

applied · Screen 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 The geoid's curvature at a point has no limit; averaged over a triangle it settles. Dashed: the curvature a geoid model contains at a point, as a fraction of the sphere's own, rising in proportion to the degree the model stops at — from 0.36 per cent at degree 360 to 36 per cent at 36,000. Solid: the geoid's contribution to the mean curvature over equilateral triangles of 1, 10, 50 and 100 km, the rms of the anomaly in each one's spherical excess divided by that excess. Each rises while the model is coarser than the triangle and settles once it is finer: the 50 km triangle reads 0.64 per cent at EGM2008's degree 2190 and 0.68 per cent at 36,000; the 1 km triangle 2.19 per cent and 27 per cent.

A triangle reads the geoid's curvature, and that one converges

The geoid's Gaussian curvature at a point has no value: its spectrum carries two more powers of the degree than the height's, and the sum never settles. A surveyor's triangle does not measure it at a point. By Gauss–Bonnet its excess is the total curvature over its area, which on the geoid is the flux of the deflection of the vertical through its three sides — a first derivative averaged along lines, and that sum converges. A fifty-kilometre triangle reads 0.64 per cent of the sphere's curvature from the geoid at EGM2008's degree and 0.68 at degree 36,000, where the curvature at a point has grown sixteenfold. The quantity nothing could measure was the one nobody measures.

impossibility · Curvature The same resolution, the grid moved, and a different answer. The same field at a fixed 18 × 9 division, with the grid slid by fractions of a cell. Nothing is lost — every cell is the same size as before and there are exactly as many of them — and the largest reported value moves over a range of 12.59 per cent. That is more than a whole halving of the resolution costs, which is 10.15 per cent on the same field. The scale effect has an excuse and this one has none.

The answer depends on the cells it was counted in

Nine essays price the cell as a shape. The number reported out of it is priced nowhere: sliding a grid without changing its resolution moves the largest reported value by 12.6 per cent, which is more than halving the resolution costs.

applied · Cells Twelve cells of the grid's pole row are one map. The pole row of the aspect grid for Robinson over a 10° square region centred at 38° N: 24 pole longitudes across, 12 turns of the third angle down, with the projection's pole on the south geographic pole. There, moving the pole in longitude and turning the third angle are the same rotation, so every cell whose longitude less its turn is the same is the same map. The twelve filled cells are one map, the lowest on the row at 1.71 times the region's best score, and they score alike to 2e-11. No two of them are neighbours. The hollow cells are the maps one 15° step round the axis, the lowest at 2.33 times the best, and on the grid as it was the copies could join only through them.

The pass was a map meeting itself

Four measurements fitted a power law to the depth of the pass that splits a near-optimal set of aspects, and found it tight — R-squared of 0.98 for Robinson, 0.996 along most of a family of maps. On the grid those aspects were searched over, a pole row holds every normal-aspect map twelve times at cells that are not neighbours, and in 28 of Robinson's 36 regions the published pass was two copies of one map meeting through the map one step round the axis. Joined as the one map they are, Robinson's curve falls to R-squared 0.008, and the sinusoidal's famous refusal turns out to be the one measurement the copies never reached.

choosing · Choosing The same labels, placed once and placed per tile. 140 stated label boxes on a 1024-pixel page cut into 256-pixel tiles. The pale boxes are placed the same way by both rules. The dark ones are placed by the tiled renderer and suppressed by the global one — labels that should have lost a collision with something in the next tile and did not, because the tile that drew them could not see it. nine of them, against 104 labels the global rule keeps.

A label belongs to no tile

Nine rungs price the tile as a piece of geometry. A label is not geometry — it is a page object placed by collision against other page objects, and collision is a global relation while a tile is rendered alone. Cut a page into tiles and 10.3 per cent of the labels are placed differently; the buffer that closes the gap is half a tile at moderate density and a whole one when it is crowded.

applied · Screen Beside a bending river, every reachable point is reached by a route that meets the bank at most three times. Everywhere within a 10 km walk of the centre, beside the stated river meandering ±2 km on a 4 km wavelength with bridges every 2.5 km, shaded by the fewest times a shortest route to it has to meet the bank — a contact being one run along the bank, however many corners it passes. 70.3% of the ground is seen from the centre across dry land. One contact reaches 28.0% more, 85% of it beyond the river and the rest in bays of the near bank hidden from the centre by a spur; two reach 1.18% and three the last 0.47%, 70% of those two beyond the river, in the lee of the far bank's bends beside the outer bridges. Nothing needs a fourth.

A layer is a meeting with the bank, not a corner turned

A reach set drawn in layers — rays from the centre, then from every place a route can leave the bank — ends in the exact answer, and how deep it must go depends on what a layer is. Counted by corners turned, the depth belongs to the polygon the bank is drawn as: five turns on a bank drawn every 400 metres, eighteen on one drawn every 50. Counted by meetings with the bank, it is three at every resolution. One meeting anywhere leaves 1.6 per cent of a ten-kilometre walk unreached, where rays from the bridges alone leave 6.4; two leave 0.47; three leave nothing. A longer walk needs more — up to six meetings at twenty kilometres across thirty-six rivers — and the bend's steepness, which prices the first layer, ranks the depth at 0.24.

paths · Reach

Copied out to sea, a coast keeps its length and loses its bays

Before 1930 a territorial sea was drawn by following the coast in all its sinuosities, and the phrase fits two constructions. Moved twelve miles seaward, a stated coast of five bays gives up 240 square kilometres against the envelope of arcs, most of it over the narrowest bay; moved along its own normals, it ties a loop over every bay. On a fractal coast charted at finer detail the copy grows by a quarter at every level, to 480 kilometres, while the envelope settles at 133.0.

practice · Boundary

Named alongside it

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

VerificationAggregationError budgetGeneralisationAreaCell systemConventionEqual-areaGeoidToleranceVector tileAliasing

All concepts