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

The thread: Computed, not quoted — page 12

Scale factors, areal ratios, angular deformations, geodesic distances and the areas of countries are all computed while the figure is drawn. None is a number recalled from a table. Essays 265 to 288 of 299.
The sea surface, against the surface heights are measured from. The stated mean dynamic topography — how far the sea stands above the geoid — drawn on Mollweide, with the eight tide gauges marked. It ranges over 2.09 metres, it is smooth, and it is what every national datum's zero is sitting on. Nothing here is a model's output: it is a stated closed form, chosen to have the observed sign structure and the observed size. What the numbers refer to

Every country's zero is a different surface

Ten rungs measure a height against a geoid and treat the geoid as one object. No national datum is on it: each is pinned to the mean sea level at one tide gauge, the sea surface stands up to two metres from the geoid, and the eight European zeros measured here spread over 462 millimetres — a step no levelling can remove and nobody's error.

 The straight line between two correct maps passes through an incorrect one. Both ends of this sequence are exact cartograms of four cities: the triangular, x first construction at s = 0 and the triangular, y first at s = 1, each meeting the density to arithmetic noise. The panels between them are the straight-line blend of the two, which is what an animation between two maps computes. The cells drawn solid have turned inside out — their signed area is negative, so the map has folded over itself there and two places on the sphere are drawn at one place on the page. Measuring distortion

A map that meets its target can fold

The flow a diffusion cartogram integrates is a diffeomorphism at every instant and cannot fold. Every discretisation of it can, and the point at which one does is a root of a quadratic — written down rather than searched for.

 Seven published numbers, twenty-eight actual ones. The correlation matrix of a seven-parameter fit to 64 common points over a region 9° across. Only the diagonal is ever published — seven standard deviations — and the twenty-one off-diagonal entries are not small: the strongest is ty against rx at 0.940. A translation and the rotation that mimics it over a small patch are very nearly the same parameter, so the fit cannot tell them apart and its errors in the two are locked together. What the numbers refer to

The parameters are not independent

Rung seven gives the seven parameters their own uncertainty and stops at seven numbers. There are twenty-eight, and the twenty-one nobody publishes are not small: a translation and the rotation that mimics it correlate at 0.94, the normal matrix has a condition number of 4 × 10¹⁶, and propagating from the diagonal alone overstates the transformation's uncertainty by up to a factor of thirty-six.

 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. What a machine does with it

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.

 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. Measuring distortion

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.

 A section through a body with a neck, and the rays that leave it twice. An equatorial section of a stated contact binary — two lobes of radius 1 centred at ±1.5, joined by a neck of radius 0.35, with the origin in the neck. The lines are rays from the origin and the marks are where each one crosses the surface. two of the 25 drawn cross more than once, so along those directions there is no such thing as "the" radius, and a longitude and a latitude do not name a place. What the numbers refer to

A ray from the centre hits the surface twice

Eleven rungs map bodies that are lumpy, triaxial and turning at a drifting rate, and every one assumes the surface is star-shaped about the centre — which is what makes a longitude and a latitude a coordinate at all. A contact binary is not: on a stated body with a neck a third of a lobe wide, 10.9 per cent of the sky has no single radius, and the shape model everybody publishes fills the neck in and adds 1.67 per cent of the volume.

 Three selection rules, and what each one keeps. Keeping one feature in ten from a stated population whose size distribution has a Pareto exponent of a half — the exponent Töpfer's law is a theorem about. Keeping the largest carries 99.99 per cent of the total size and inflates the median feature by a factor of 95. A random sample keeps the median to 1.068 and carries 5.0 per cent of the total. The two rules are right about different things and there is no rule that is right about both, because the total lives in the tail and the median does not. What a machine does with it

Which features survive is not a sample

The rung below answers how many features a scale can carry and treats the population as a number. Which ones survive is a different question: keeping one feature in ten carries 99.99 per cent of the total length and inflates the median feature by a factor of 95, and the shape of the size distribution survives both exactly.

 How far the two routes end up apart. The distance between a line simplified directly at the final tolerance and the same line simplified through two intermediate products, as a multiple of the final tolerance, with a stated extra step applied to each intermediate. With nothing in between the two are the same line to the last bit, because Douglas–Peucker's outputs are nested. Rounding the intermediate to half the tolerance, smoothing it for legibility, or running it through a moving average each break that, and the last of them puts the final product 0.142 away from where a direct route would have put it — nine times the tolerance the product is published under. What a machine does with it

Two routes to one scale

A national series is cascaded — the million is derived from the quarter-million, which was derived from the fifty — and the folklore is that the errors accumulate. They do not: Douglas–Peucker and Visvalingam both cascade to the same line the direct route produces, bit for bit, because both output a sublevel set of a per-vertex number. What breaks it is anything else in the chain, and a moving average puts the product nine tolerances away.

 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. Measuring distortion

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.

 Two maps of one density: one costs nothing and one costs forty-four degrees. Both drawings meet the same areal request — an exponential ramp of contrast 79.8 to one, whose logarithm is harmonic — to arithmetic noise. The first is a conformal map written down in closed form, the conformal map that meets an exponential ramp, log-harmonic, whose angular deformation is 2.4e-8 degrees. The second is the triangular construction the previous rungs use, at 43.8° on the same request. A grid of squares is drawn through each: the first keeps every angle and the second does not. Measuring distortion

The cheapest map that meets its areas

An earlier essay bracketed a cartogram's least cost between a construction charging eighty degrees and a bound valid only for symmetric densities, and recorded the gap as a shortfall. One request settles it: a density of contrast eighty whose least cost is exactly zero, met by a map written down in closed form, while the standard construction charges 43.8° for it.

 One rule, applied once, to one dataset. Two features 3 metres apart on the ground, and a snapping tolerance of 5 map units on Web Mercator. The tolerance's reach is the curve; the pair's separation is the line. They merge everywhere below about 52.8° and stay separate above it — the same features, the same rule, the same run. A pipeline that cleans a global dataset in one pass produces a topology that changes at a latitude nobody chose and nothing records. What a machine does with it

A tolerance in map units is not a tolerance

A snapping tolerance is a number, and the number is in whatever units the file is in. Five map units on Web Mercator is 4.97 metres of ground at the equator and 0.87 at eighty degrees — so a rule that merges two features three metres apart merges them everywhere below 52.8° north and refuses everywhere above it, in one pass, over one dataset, with nothing recording where the boundary is.

 Millimetres of horizontal move per kilometre of assumed height. How far the transformed latitude and longitude move when the height fed into a datum transformation changes by one kilometre, for four published transformations at five places. It runs from 7.7 to 121 millimetres per kilometre. A latitude and a longitude do not carry a height, so a two-dimensional coordinate cannot be transformed until somebody supplies one that is not in it. What the numbers refer to

The height a coordinate does not carry

Ten rungs treat a datum shift as a map from one pair of angles to another. It is not one: the transformation runs through Cartesian coordinates, so the answer depends on the height — by 7.7 to 121 millimetres per kilometre depending on the datum and the place, which puts Lhasa 432 millimetres from where a height of zero would have said.

 Two conventions for one indicatrix field on Mercator. The left panel is the convention nearly every published field uses: every ellipse drawn at the same area on the paper, so the picture carries the shape and throws the size away. The right panel draws the image of the same ground circle at every place, so the drawn area IS the areal scale factor. On Mercator the axis ratio spans 1.0000 and the areal factor spans 14.93, so the left panel shows nothing the other one shows. Neither caption states which is which, on any published map this collection has found. Measuring distortion

The ellipses are a sample, drawn at a size somebody chose

Thirteen essays measure with the indicatrix and none audits it as an instrument. A published field has a gauge nobody states and a placement nobody states: on Mercator the standard convention draws twenty-five identical circles while the areal factor runs over a factor of 14.9, and the average a reader takes off any of these fields is between 22 and 64 per cent too high.

 Waiting longer buys almost nothing under the spectrum that is actually there. The scatter of block means against the block length, for the three spectra. White noise falls with a slope of -0.484 — the inverse square root everybody assumes. Flicker falls with a slope of -0.111, so a hundredfold longer average buys a factor of 1.67 rather than ten. A random walk is very nearly flat. Measuring distortion

The error that does not average down

Five rungs give a coordinate a width, and all five assume the observations it was averaged from are independent. They are not. Under the noise spectrum every published analysis of a position time series reports, the standard error falls as the eleventh root rather than the square root — so a factor of ten costs a hundred observations if the noise is white and a thousand million if it is not.

 The bias is σ² over the length, over two decades. The amount by which a measured baseline is longer than the true one, against its length, for a twenty-millimetre error on each end. The line is the second-order prediction σ²/d. The measured bias times the length is constant to a factor of 1.0014 across the whole range, and it sits 8.2 per cent below the prediction — which is the fourth-order term the expansion drops. The estimate is antithetic, so the first-order scatter cancels exactly and a bias of thirty-seven microns is measured at a t-statistic of 349. Measuring distortion

A length measured from noisy points is too long

A distance is a square root, a square root is concave, and the average of the distances is not the distance between the averages. The gap is a bias with one sign: 37 microns on a ten-metre baseline with twenty-millimetre marks, following σ² over the length across two decades, and it adds rather than cancelling — so the same boundary is 1.5 parts per million longer when it is measured in more pieces.

 Two compromises along one path, and they do not agree. The angular deformation and the flexion of every blend between Mercator and Lambert cylindrical, each divided by the straight line between the two parents' own values. A value of one means the blend is exactly the average of the two errors and has bought nothing. The angular deformation dips to 0.649 at a weight of 0.13; the flexion dips only to 0.955, and it does so at 0.50. Anybody choosing a compromise is choosing a weight, and the two orders want different ones. Measuring distortion

The best compromise for angle is not the best for bending

Every compromise projection in the library is an average of two others, and averaging is a first-order operation — so the second derivative was never part of the bargain. Swept along five ordinary blend paths, the weight that minimises angular deformation and the weight that minimises flexion are between a quarter and a half of the axis apart, and how much a compromise buys at one order predicts nothing about the other.

 The same field, the same statistic, two pages. A single northern concentration — v = 10 + 90 exp(−d²/(25°)²), d the distance from 20°E 55°N — drawn twice, with the shading rule and the data identical. The ground's own area-weighted mean is 14.009. A reader weighting by the area actually on the page takes 17.490 off Mercator, which is +24.85%, and 14.009 off Gall–Peters, which is +0.00%. Neither map has misdrawn a single cell. Measuring distortion

A choropleth is read by area

Every cartography course states the rule — use an equal-area projection for a thematic map — and states it as advice. It is a theorem, and it has a residual: the error a page puts into a reading is exactly the covariance of the value with the areal factor, which is 24.85 per cent for a northern concentration read off Mercator and 0.00 per cent for the same field read off a map that spreads area by 7.7 to one.

 The same totals, drawn as symbols, on two pages. A band field, largest in the mid latitudes carried by each region and drawn as a circle whose area is proportional to the total. Every symbol is right: a total is a total wherever it is drawn, and the two maps carry identical numbers. What differs is the ground under each symbol. On Mercator the symbol at 60° sits on a region drawn 4.0 times larger than it would be on an equal-area sheet, so the density a reader forms — symbol against region — is out by that factor. Measuring distortion

A symbol has a size on the page and an area on the ground

A proportional symbol is right as a total wherever it is drawn — a count is a count. Read against the region beneath it, which is how a reader forms a density, it is wrong by exactly the reciprocal of the areal factor: 0.083 at 73° north on Mercator, a factor of twelve, with the correction available as one multiplication that no atlas makes.

 The same count, scattered two ways. 22 dots in every cell of a 30-cell covering, drawn on Mercator. The upper panel places them uniformly on the GROUND — uniform in longitude and in the sine of latitude, which is what uniform on a sphere means — and the lower places them uniformly on the PAGE, which is what a drawing routine handed a polygon does. Both panels carry exactly the same number of dots in exactly the same regions, so both are honest as totals. They are different pictures, and a reader reads a dot map by density. Measuring distortion

A dot map's density is partly the projection's

A dot map carries the right number of dots in every region whichever way it is drawn, so it is honest as a total under both placements. It cannot be honest as a density under both: ground on a uniform field reads 0.099 of its equatorial density at 72° north on Mercator, and scattering inside the polygon on the page moves 64.3 per cent of a cell's dots into its northern half without one of them leaving the cell.

 One field, one classifier, two sets of breaks. A field that varies with latitude put into 5 classes by area-weighted quantiles, on Mercator. The upper panel weights each region by its ground area and the lower by the area it occupies on this page, which is what a classifier handed projected geometry does. 60 of 150 regions land in a different class, marked in the lower panel. The data has not changed and neither has the number of classes. Measuring distortion

The class breaks were computed on the page

The three rungs below price what a reader does with a finished map. A classifier is software, it runs on the geometry it has, and the geometry it has is projected: a five-class quantile classification of one stated field puts half of the three hundred and eighty-four regions in a different colour on Mercator, and 87.5 per cent of them at nine classes.

 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. Measuring distortion

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.

 The areal factor over 0° to 60° north, and the points it is measured at. Mercator's areal factor shaded over 0° to 60° north, from 1.0001 to 3.8473, with the 12 × 12 grid of cell centres a regional measurement uses drawn on top of it. The cross marks where the quantity is actually largest, found by a search that is allowed to leave the sample; the ring marks the largest value the sample contains. The grid's answer is 3.4639 and the real one is 4.0000, short by 13.40% — and the reason is visible in the picture, because no cell centre is ever on an edge. Measuring distortion

The worst point is not on the grid

Every maximum distortion this collection has printed is a maximum over a sample, and a maximum over a sample is a lower bound. On Mercator over a sixty-degree band a twelve-by-twelve grid reports 3.464 where the answer is exactly 4, and the shortfall does not go away with refinement so much as decay at a rate that says where the extreme is hiding.

 One of these averages exists. Mercator's area-weighted mean areal factor, and its Kavrayskiy number, against how close the sampled band comes to the pole. The first is artanh(sin Φ)/sin Φ in closed form and has no limit: it passes 7.04 at a tenth of a degree from the pole and keeps going, gaining a fixed amount every time the remaining gap is halved. The second settles by 85° and does not move again. Both are published as summary distortion figures for the same map. Measuring distortion

A mean that does not exist can still be printed

Mercator's area-weighted mean areal factor is artanh(sin Φ)/sin Φ, and it has no limit. A sampler asked for it returns the logarithm of its own sample count plus 1.512 — measured slope 1.001 against ln n — so the number is a property of the person who computed it. The Kavrayskiy number for the same map over the same sphere settles at 0.52124 and is a number.

 Three ways to cover a sphere with 900 points. The three samplers this collection's numbers are computed with, each with about 900 points, drawn on the Mollweide projection so that equal areas on the sphere are equal areas on the page and the crowding is the samplers' rather than the map's. The graticule piles points at the poles; the equal-area rings space them evenly by area and unevenly by distance; the Fibonacci lattice trades a little of each. None of them is equal-area, and no finite set is. Measuring distortion

There is no equal-area lattice on a sphere

Every number in this collection is an average over a point set, and the three point sets available all fail to be equal-area in different ways. The equal-area ring sampler this site has used since its early essays is the one whose outermost ring sits half a step inside the rim — which is how a measured scale spread once came in 3.42 parts in a thousand below a proved bound.

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