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

Projection identification — where it appears

Recovering which projection drew a map from the map itself, by fitting every candidate to the graticule and ranking the residuals. Its residual is not labelled: a wrong datum and a wrong projection produce the same symptom, and they separate by how the residual grows with the region.

Named by 12 essays across one field — each of them below, with the objects they name alongside it.

Every candidate fitted to one map's graticule, ranked by what is left over. The map is drawn in Conformal conic over a region 40° tall centred at 45° north, and the projection is not told to the fit. Each candidate is evaluated at the same 121 graticule crossings, its own parameters are searched, the best plane similarity between its output and the picture is removed, and what remains is drawn as a proportion of the map's own width on a logarithmic scale. Conformal conic fits to 1.3e-10, which is the arithmetic's floor; the next candidate is 3.7e+7 times worse.

A map does not say what it is

Every map on this site is one the site drew, from a projection it chose. Every map a reader has ever used is the other kind — a picture whose projection is a sentence in a corner, a legend, or nothing at all. The graticule is enough to recover it: fit every candidate to the crossings and rank what is left over.

wrong · Identify How large a map has to be before a wrong projection stops fitting it. Three rivals fitted to a Mercator graticule centred at 45° north, over regions from 1° to 70° of half-extent. The residual is the shape difference alone, with the best scale, rotation and offset removed, and both axes are logarithmic. The horizontal rule is a fifth of a per cent of the map's width — about the width of a drawn line on a printed sheet — and where a curve is below it, no measurement of that map can tell the two projections apart, however carefully it is made.

Two projections that cannot be told apart

Over a small enough region every projection is the same picture, so the question is how small. The answer is not one number: two conformal projections need twelve degrees of extent before their graticules can be separated, and two projections with different anisotropy separate below one.

wrong · Identify The half-extent at which each rival stops fitting a Gall–Peters graticule. For each candidate, the size of region at which its best fit to a Gall–Peters map first leaves a residual of a fifth of a per cent of the map's width. Below that size the two are the same picture. The numbers are half-extents in degrees of latitude, at 20° north, with the plane affine transformation removed; a bar at 90° is a rival that never separates at all within the range searched.

What a careless copy hides

A photocopier that stretches one axis is a nuisance to remove before a projection can be identified. Removing it costs more than it looks: an affine fit absorbs the entire difference between the cylindrical equal-area projections, so Gall-Peters and Behrmann become the same picture at any size, to sixteen decimal places.

wrong · Identify The same ranking, with the right answer removed from the library. A map drawn in Mercator, fitted by every candidate except Mercator. Something still wins: Conformal conic, by a factor of 1.64 over the runner-up, leaving 0.4 per cent of the map's width unexplained. With Mercator in the library the winner's margin is 1.2e+13. The ranking always produces a name; what tells the two situations apart is how far ahead the name is.

When the answer is not in the library

Fitting twenty candidates to a map and ranking the residuals always produces a winner, which makes it a ceremony unless it can also produce a refusal. Held out of its own library, a Mercator map is named as a conformal conic, leaving 0.4 per cent of the map's width unexplained — and the quantity that tells the two situations apart is not the residual but the margin, which is 1.6 when the truth is absent and 10¹³ when it is present.

wrong · Identify What each configuration can see, and what it cannot. The smallest eigenvalue of the fit's own normal matrix — how much the residual changes for a unit move in the worst direction of parameter space — for three parameterised candidates against six configurations of the same size. A zero is not a hard fit: it is a direction the control points cannot see at all, so every value of the parameter along it gives an identical residual. 2 of 18 are at the floor of double precision, and they are not the ones a reader would guess. Where none is, the spread between the best and worst arrangement is still Infinity at a fixed point count.

Where the control points are

Five rungs fit a library to a map and ask how much to trust the winner. None asks whether the parameters are recoverable at all. On control points along one parallel an equirectangular's standard parallel is not merely hard to find — it is invisible, exactly, and five hundred and twelve points on the same parallel are as blind as eight.

wrong · Identify A coast that agrees with its graticule, except where it was copied. The outline of Japan on a map whose graticule is a conformal conic projection. 30 per cent of the coast, drawn in the second ink, was copied from a sinusoidal sheet of the same ground and fitted into place so that its two ends meet the coast either side, the dots. The dashed line is the coast the graticule predicts. Everywhere else the drawn coast lies on it exactly; along the copied stretch it departs by up to 9.4 thousandths of the map's width. Nothing in the graticule shows it.

A compiled map agrees with its graticule except where it was copied

A map compiled from two sheets has no single projection behind it, and its graticule cannot say so. Its coast can — not as a residual, which is a number the size of a projection error, but as a profile: the drawn coast lies exactly on the one its graticule predicts, then departs from a seam, and returns at another. The seams are found to within one point in 720, and the stretch between them names the projection it was copied from with a margin of 3.6 × 10¹¹.

wrong · Identify The same copied stretch, dropped in rigidly and eased into place. One coast drawn twice. In the first ink the copied stretch is moved as a rigid, scaled piece and meets the rest of the coast at two corners — the dots. In the second it is pinned at the same two points and then eased along 30 per cent of the outline either side of each of them, which is what a compiler fitting a sheet by eye does. The dashed line is the coast the graticule predicts. The eased version departs from it by 6.1 thousandths of the map's width against the rigid one's 9.4: easing moves the departure about, and removes very little of it.

A copy eased into place loses its seams before its source

A stretch of coast lifted from another sheet and dropped in rigidly leaves two sharp corners where it joins, and both of them are findable. A compiler does not work that way: they pin the ends and ease the middle until it looks continuous. Easing halves the corner within five per cent of the outline and the seams survive it — until forty per cent, where the located seam jumps from four per cent of the coast out to twenty. The source projection survives twice as far, to seventy-five. The piece goes on naming where it came from after the method has lost track of which piece it is.

wrong · Identify Charged ln N a number, the count comes out right; charged two, it takes all it is offered. Two information criteria over the same seven segmentations, each shown above its own minimum. Both add the log of the residual to a charge for every number the model spends — four for each piece's placement and one for each seam's position. BIC charges ln N, 6.6 for 720 coordinates, and is least at 2 seams, the number made. AIC charges two, and is least at 6, the most it was offered: each extra seam buys more than two units of fit because its position and its projection are both chosen by search, and a charge for the numbers alone does not pay for the search.

A residual cannot count the pieces of a map

A compiled map does not say how many sheets went into it, and trying every count and keeping the best fit cannot tell. Seven pieces always fit better than two, because they contain them. A criterion that charges ln N for each number a piece spends recovers the true count in every trial, for one copied stretch and for two, until the digitising error reaches a thousandth of the map. AIC charges two and takes every piece it is offered. And an eased copy defeats the right criterion the other way: it is not a projection, so BIC buys pieces to explain it, three seams, then four, then six.

wrong · Identify With the drift's length known, the count comes back. A compiled coast of 360 points traced with a drifting error — each point's error mostly the last one's, over a correlation length of 18 points, three ten-thousandths of the map in size — and the number of seams BIC chooses once the error is whitened at an assumed length, eight tracings each. No copy (0 seams made): 0/8 right at none, 1/8 right at 2, 8/8 right at 6, 8/8 right at 12, 8/8 right at 18, 8/8 right at 30, 0/8 right at 60; one copy (2 seams made): 0/8 right at none, 6/8 right at 2, 8/8 right at 6, 8/8 right at 12, 8/8 right at 18, 7/8 right at 30, 0/8 right at 60; two copies (4 seams made): 1/8 right at none, 7/8 right at 2, 8/8 right at 6, 8/8 right at 12, 8/8 right at 18, 5/8 right at 30, 0/8 right at 60. Unwhitened the criterion buys pieces for the drift in every compilation; whitened at 6, 12, 18 points it counts every compilation right; assumed 3.3 times too long, it takes the most it is offered.

A second tracing tells a drifting hand from a copied stretch

A hand tracing a coast drifts rather than jitters, and a drifting error is a smooth departure no projection explains — so the criterion that counted copied stretches correctly under independent noise counts four or five seams where two were made, and three or four on a coast with no copy at all. Undo the drift before fitting and every count comes back, for no copy, one and two, in every tracing. That needs the drift's correlation length from outside the coast, known to within a factor of three on the short side and much less on the long, and a second tracing of the same coast supplies it: the difference between two tracings holds nothing but the two errors, and read that way the count is right in 22 tracings of 24, and never too high.

wrong · Identify Given an easing to fit, the criterion counts an eased copy as one copy until the two easings run well into each other. One copied stretch, eased into place over a stated share of the coast either side of each seam, traced to a ten-thousandth of the map. Hollow: the rigid count, offered up to six seams — two for a rigid copy, then 4, 5, 6, 6, 6, 6, 6, 6 from 5.0% of easing. Filled: the count when each seam may be eased over a width the fit finds and pays for, offered up to four — two at every easing to 17.5%, then 4, 4. The dotted line is half the copy's length, 15% of the coast: past it the two easings overlap, and the copy is nowhere drawn as itself.

An easing is a seam with a width, and a drifting hand hides the width

A copy eased into a compiled coast is not a projection, so a count that explains the coast in rigid pieces buys four, five or six seams for one copy. Give each seam a width — the copy blended into the host over that width, the same linear fit at every width, the width charged as one more number — and the count is two seams at every easing up to 17.5 per cent of the coast, and the width it reports is the one drawn, wherever the copy departs from the host by many times the digitising error. Where it departs by seven, the width is a guess. On a tracing made by a drifting hand the easing disappears: whitened for the drift, the rigid count already calls two seams in every tracing, because an easing as long as the drift is a stretch of drift, and its width can no longer be read until it is twice the drift's length.

wrong · Identify Averaging divides the hand's wander and leaves its habit: a shared part sets a floor no stack goes below. The drift left in the mean of N tracings, as a share of one tracing's, when a stated part of the hand's variance is a habit shared by every tracing: none (thick), a quarter, and half. With none it falls as one over the square root of N, to 25 per cent at sixteen; a quarter shared stops it at 50 per cent, half at 71. Dotted: the drift at which an independent hand's stacks read the strong seam's easing (two tracings, 71 per cent) and the weak seam's (sixteen, 25).

Averaging tracings reads the easing, until the hand repeats itself

One tracing by a drifting hand cannot show how far a copy was eased, because an easing as long as the drift is a stretch of drift. Average several tracings and the easing stays while the drift shrinks: two tracings read the easing's width at the seam where the copy departs strongly, sixteen read it at the seam where it barely departs, and as the drift falls the rigid count starts buying pieces for the easing again while the eased count stays at two. All of that assumes each tracing drifts afresh. A hand that repeats part of its drift in every tracing leaves that part in the mean, where it sets a floor no number of tracings goes below; and a digitiser who learns the hand from the spread between tracings measures only the part that averages away, and charges the mean too little, so that both counts buy every piece they are offered.

wrong · Identify Sixteen tracings by four hands leave half the drift that sixteen by one hand do, unless the hands share a habit too. The drift left in the mean of sixteen tracings of one sheet, as a share of one tracing's, measured over 120 sheets, for sixteen tracings by one hand, eight each by two, four each by four, two each by eight and one each by sixteen. When half of each hand's drift is its own habit (darker): 1 × 16 74, 2 × 8 54, 4 × 4 40, 8 × 2 31, 16 × 1 25 per cent. When a quarter is the hand's habit and a quarter a habit every hand in the office shares (paler): 1 × 16 73, 2 × 8 63, 4 × 4 58, 8 × 2 56, 16 × 1 54. Dotted: 71 and 25 per cent, the drift at which an independent hand's means read the strong seam's easing and the weak seam's.

The spread between hands charges the mean, and an office's habit hides in the sheet

One hand's tracings cannot see the part of its drift it repeats, so a mean of sixteen by one hand is charged for a third of the error it carries. Sixteen tracings by four hands carry half the error of sixteen by one, and the spread between the hands' means, divided by the number of hands, charges that mean correctly to within a fifth on every sheet from three hands on — without ever estimating how much of a hand is habit. The same spread reads a hand's share of habit from two hands of two. What it cannot read is a habit every hand in the office shares: tracings of a sheet with one are tracings of a different sheet without it, and the charge they give falls further short the more hands and tracings are added.

wrong · Identify

Named alongside it

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

ResidualLeast-squaresCompilationEstimatorSeamVerificationIdentifiabilitySimilaritySimilarity transformationCandidateDegrees of freedomGraticule

All concepts