Two projections that cannot be told apart

A map does not say what it is recovers a projection from its graticule and wins by a margin of thirty-seven million. That margin was measured over a region forty degrees across, and it is not a property of the method.

Shrink the region and every projection becomes the same picture. That is not a limitation of fitting: it is a restatement of the fact this whole collection begins with — how small is flat enough measures the size at which the sphere stops mattering, and below it every map of the region is a plane transformation of every other, which is exactly what the fit removes.

So the question with an answer is: how small? And the answer is a table rather than a number.

What separability means here #

A pair is separable over a region when the best fit of one to the other’s graticule leaves a residual larger than a drawn line — a fifth of a per cent of the map’s own width, which on a hundred-millimetre map is two tenths of a millimetre.

That threshold is a convention and is stated as one. A finer line moves every extent in this essay inward and a coarser one moves them out, in a way that is easy to compute from the curves themselves: they are power laws, so halving the threshold moves an extent by a factor that is the reciprocal of the exponent. What is not conventional is the shape of the curves it cuts, and the shape is what this rung is about.

Twelve degrees, or under one #

The measured extents split the library into two groups with almost nothing in between:

• Mercator against the stereographic: separable at a half-extent of 12.13°;
• Mercator against the conformal conic: 11.71°;
• the stereographic against the conformal conic: 10.20°;
• Mercator against Miller’s: 0.75°;
• Mercator against the plate carrée: 1.50°;
• Mollweide against Eckert IV: 0.75°;
• Gall–Peters against Behrmann: 0.75°.

The first three pairs are conformal at both ends. The rest are not. Two conformal projections need a region ten to twelve degrees across before their graticules can be separated at all, and every other pair separates over a region a surveyor would call local.

Why conformality is what decides it #

The reason is the same one that runs through this collection’s treatment of Tissot’s construction, and it is about the order at which two maps first differ.

Fitting a similarity to a graticule removes everything of first order: a scale, a rotation and an offset. What is left is whatever the two projections do differently beyond that.

Two projections with different anisotropy — one stretching north–south relative to east–west, the other not — differ at first order, in the derivative itself. A similarity cannot absorb that, because a similarity is isotropic. The residual is therefore of the same size as the region, and shrinking the region shrinks the map and the residual together: the proportion stays put.

Two conformal projections have no anisotropy at all, at any point, by definition. Their derivatives differ only by a scale and a rotation — which a similarity absorbs exactly — so they agree to first order everywhere, and what is left over is the second derivative and beyond.

The measurement confirms it with an exponent. Fitting the relative residual against the half-extent as a power law:

• Mercator against the stereographic: exponent 1.90;
• Mercator against the conformal conic: 0.93;
• Mercator against Miller’s: 0.02;
• Gall–Peters against Behrmann: −0.04.

An exponent of zero says the proportion does not shrink with the region: those pairs differ at first order and are separable at any size a line can be drawn at. An exponent of one says the difference is one order higher. An exponent of two says the two maps agree to second order as well, which is what a conformal pair with a free centre parameter can arrange.

The consequence for a real map #

The practical reading is uncomfortable and worth stating plainly. A conformal map of a country the size of Switzerland cannot be identified from its graticule, because every conformal projection fits it to within a drawn line — and conformal projections are exactly the ones national grids use, for the reasons a grid is a conformal map sets out.

That is not a failure of the fitting; the information is not in the picture. Two conformal maps of a small region differ by a third-order term, and a third-order term over ten degrees is smaller than the ink.

What can be identified from a small region is a projection that is not conformal, because the anisotropy shows immediately. So the method’s reach is the opposite of what one would guess: it is best on the projections that are worst behaved, and blind exactly where the mapping agencies concentrate.

Density does not help, and cannot #

The first thing anybody suggests is to draw the graticule more finely, and it is the wrong instinct. The residual converges: from 7.84 × 10⁻³ at 25 crossings to 4.90 × 10⁻³ at 1,089, heading for 4.456 × 10⁻³.

It converges because the quantity being measured is a shape difference over a region, not a signal being sampled. Sampling the same shape more finely does not reveal more shape. What the density does affect is the arithmetic — a root-mean-square over 25 points is a coarse estimate of a root-mean-square over the region — and each halving of the step closes about half the gap, which is first-order convergence to a limit that belongs to the region alone.

What decides identifiability is the extent, and only the extent. The rest is estimation.

Where the region sits, and not only how large it is #

Extent is what decides identifiability, and where the region sits decides the constant in front of it.

Near a projection’s own standard parallel — the equator, for Mercator — the map is close to a similarity of the ground, so every candidate that is roughly right is very nearly right, and the residuals fall together. Far from it, the projection’s own distortion is large and a rival has more to fail to reproduce.

That has a consequence worth stating for anybody trying this on a real sheet: a map of an equatorial region is harder to identify than a map of the same size at high latitude. The information a fit uses is the projection’s own departure from a similarity, which is exactly the quantity every other essay on this site calls distortion — so the projections that are least distorted over a region are the ones hardest to name from a picture of it.

The indicatrix is the same statement at a point #

The whole finding has a one-point version, and putting it beside the region version makes the mechanism plain.

Tissot’s indicatrix describes the first derivative of a map at a point. For any conformal projection it is a circle, and for two conformal projections it is the same circle once each map’s overall scale is taken out. So the indicatrix — the site’s central instrument — carries no information at all about which conformal projection is which.

The indicatrix is a limit measures how a finite circle departs from that description, and finds the departure first order in the radius. That departure is exactly the information this rung’s fit is using: the difference between two conformal maps is invisible at a point, appears in how a finite region’s shape departs from the point description, and grows with the region.

A projection is identified by what its indicatrix does not say.

What was computed, and how #

For each pair, a bisection on the region’s half-extent.

The map is generated in the true projection over a box of the stated half-extent, at a graticule step proportional to the extent so that the number of crossings is roughly constant. The rival is fitted with its own parameters searched. The residual is divided by the map’s own width, and the bisection finds where that proportion crosses the threshold.

The power-law exponents come from fitting the same proportion against the half-extent at five sizes, in logarithms, which is the same fit this collection uses for every convergence order it reports.

One structural detail decides several of the numbers. Some candidates cannot show the region at all — a gnomonic projection asked for a region spanning more than a hemisphere, an azimuthal asked for its own antipode — and such a candidate has to return an honest infinity rather than a number. The first version of the search kept a running best without checking that any candidate had produced a finite score, and then read a parameter off a result that did not exist.

Where the model stops #

The threshold is a convention. A fifth of a per cent of the map’s width is a drawn line on a printed sheet; a digital map at high zoom has a much finer effective line width, and a pair that is inseparable on paper may be separable on a screen. All the extents scale with that choice, and none of the exponents do — which is why the exponents are the transferable part.

The regions are boxes centred at 45° north. A pair’s separability depends on where the region is: near a projection’s own standard parallel the two agree better than away from it, and a pair measured at the equator gives different numbers.

The observations are exact. A real reading of a map has a position error, which puts a floor under every residual and pushes every extent in this essay outward. The floor would be the first thing a working implementation had to measure, and the second essay of this ladder would then have a second threshold in it.

And the group of allowed plane transformations is fixed. Everything here removes a similarity. Allowing more — which is what a careless reproduction forces — changes the answer entirely, and for one family of projections it changes it to never.

What the table is good for #

Reading the extents as a warning is only half of what they are. The other half is a design rule for anybody who wants a map to be identifiable, and it inverts every number above.

Print the graticule over as much extent as the sheet allows. A world map is identifiable by any of these methods; a city plan is not, whatever else is on it. A sheet whose graticule covers ten degrees of latitude is at the edge for a conformal projection and comfortable for anything else.

Label the parallels far from the standard one. The information is in the projection’s departure from a similarity, so the crossings that carry it are the ones where the map is most distorted — and a sheet that crops away its own high-latitude corners has thrown away exactly the evidence.

Or state the projection, which is what a metadata field is for and what the five declarations a coordinate needs is about. The geometry can supply one of the five and only over a large enough region; the other four have to be written down whatever the extent.

That is the practical shape of the finding, and it is a little unusual for this collection: most of the essays here measure something a cartographer cannot change, and this one measures something that is entirely a matter of what gets printed.

The two thresholds a real reading has #

Everything above assumes the crossings are read exactly, and a real reading has an error of its own — a fraction of a millimetre on a printed sheet, plus whatever the paper has done since it was printed.

That puts a second threshold under the first, and the two behave differently. The line-width threshold is a property of the drawing and scales with the map’s own size, so it stays put as a proportion. The reading threshold is a property of the instrument and shrinks as a proportion when the sheet is larger, because the same tenth of a millimetre is a smaller fraction of a bigger map.

A working implementation therefore has to compare its residual against both, and the binding one changes with the sheet. On a small sheet the reading dominates and the extents in this essay are optimistic; on a large one the shape difference dominates and they are the right numbers. Neither threshold is a property of the projections, which is why the exponents rather than the extents are what this rung offers to carry elsewhere.

The generalisation #

The structure is a general fact about fitting and it is worth stating in the form this case makes concrete. Two models that agree to order n over a region of size h differ by a term of order hⁿ⁺¹, so the size of the data decides which models can be told apart — and no amount of precision substitutes for extent.

That is why a calibration measured over a narrow range extrapolates badly, why two rate laws that agree at small times need long experiments to separate, and why a fit’s ability to choose between models is a property of the design rather than of the algorithm. The cartographic version is unusually clean because the order at which two projections agree is a property with a name: conformality means agreeing to first order with every other conformal map, everywhere.

Who found it, and when #

That every map is locally a similarity is the content of differentiability and is as old as calculus. Its cartographic form — that a small enough region is described equally well by any projection — is the everyday justification for local plane surveying and is what makes a plane survey legitimate up to a stated radius.

The identifiability question is younger and belongs to the georeferencing literature, where it appears as the practical difficulty that fitting several projections to a small scanned map produces indistinguishable residuals. What this rung adds is the reason, in the form of an exponent that says which pairs are hopeless and which are merely hard.

The identification case makes the point concrete in a way that is easy to state and easy to forget. A researcher holding a small historical sheet and a library of candidate projections has a fixed quantity of information, set by how much of the world the sheet covers, and no analysis increases it. Scanning at a higher resolution, digitising more control points, or fitting more carefully all improve the precision of the residuals and leave the separation between candidates exactly where it was — because the separation is a difference between two model predictions, and over that extent the two models predict nearly the same thing.

The generalisation is worth restating in the form a practitioner meets it, since the exponent is doing all the work. Two models that agree to order $n$ are separated by data spanning a range where the order-$n{+}1$ term exceeds the noise, and that is a statement about the design of the observation rather than about the analysis of it. No amount of care with the fitting recovers a distinction the data does not contain; more points in the same small region buy precision on the parameters and nothing on the choice between models. What buys the choice is extent — a wider range, a longer time, a bigger map — and the exponent says how much extent is needed.

Where the ladder goes next #

Everything here removes a similarity, which is what a photocopier does. A reproduction that stretches one axis is an affine transformation, and allowing one changes the whole table: the pairs with exponent zero are exactly the ones an affine fit removes entirely, and one family of projections becomes unidentifiable at any size at all. That is what a careless copy hides.