What a careless copy hides

The two rungs below this one fit a projection to a graticule after removing a similarity — a uniform scale, a rotation and an offset. That is what a photocopier and a crop do to a picture, and taking it out is obviously right: none of it is a property of the projection.

Real reproductions are not always so well behaved. A sheet scanned at different resolutions horizontally and vertically, a figure squeezed to fit a column, a photograph taken at an angle and rectified: all of them stretch one axis relative to the other, and none of that is a property of the projection either.

So the fit should allow an affine transformation — independent horizontal and vertical scales, and a shear. Allowing one has a cost that is not obvious and is much larger than the extra freedom suggests.

Fitting five pairs twice settles the cost. Allowing only a scale, a rotation and an offset separates every pair; allowing independent scales and a shear lands the first three on the arithmetic’s floor, which means the whole difference has been absorbed, and merely helps the other two.

The family that is one map #

Every cylindrical equal-area projection has the same form: x = λ cos φ₀, y = sin φ / cos φ₀, for a standard parallel φ₀. Lambert’s original takes φ₀ = 0; Behrmann takes 30°; Gall–Peters takes 45°; Trystan Edwards, Hobo–Dyer and half a dozen others take values in between, and each is presented as a distinct projection with a name and an advocate.

Written down, the difference between two members is one number:

$$(x, y) \;\longmapsto\; (cx,\; y/c), \qquad c = \frac{\cos\varphi_0'}{\cos\varphi_0}$$

which is an anisotropic scaling — an affine transformation. Every member of the family is an affine image of every other member, exactly, with no approximation and at every point of the map.

So an affine fit cannot tell them apart. Not approximately, not over a small region: not at all. Fitted to a Gall–Peters graticule, the Behrmann projection leaves a residual of 7 × 10⁻¹⁶ of the map’s width, which is double precision’s floor. Under a similarity fit the same pair leaves 6 × 10⁻², which is six per cent and unmissable.

What each transformation absorbs #

Putting the three groups in order makes the pattern plain.

A translation absorbs where the origin is, which is a fact about the crop.

A similarity additionally absorbs the reproduction’s overall scale and rotation, which are facts about the paper and the scanner. What survives is everything about the projection’s shape, including its anisotropy.

An affine transformation additionally absorbs one anisotropy and one shear — and the anisotropy is not always a fact about the reproduction. For the cylindrical equal-area family it is the projection’s only parameter.

The general rule is one line. A fit can only recover what the allowed nuisance transformations do not already contain, and any parameter of a projection that acts on the map the way a nuisance does is unrecoverable.

Which pairs the extra freedom merely helps #

The extra freedom always lowers a residual — six unknowns fit at least as well as four — so a lower residual under an affine fit is not evidence of anything by itself. What distinguishes the family is that its residual falls to zero rather than merely falling:

• Gall–Peters against Behrmann: 6.0 × 10⁻² under a similarity, 7.1 × 10⁻¹⁶ under an affine fit;
• Gall–Peters against Lambert’s: 1.0 × 10⁻¹ against 2.6 × 10⁻¹⁶;
• Mollweide against the sinusoidal: 2.2 × 10⁻² against 1.4 × 10⁻²;
• Mercator against the stereographic: 1.7 × 10⁻² against 1.6 × 10⁻².

The last two pairs are not affine images of one another and the extra parameters buy 40 per cent and 6 per cent respectively. The first two are, and the extra parameters buy everything.

A ratio of 10¹⁴ between the two fits is the signature of an exact relationship, and it is worth more than the individual numbers: it says that what has been removed is not a small residual but a structural identity.

This is the same fact the Peters argument was about #

Three members of the family at the same point have an areal factor of exactly one and are conformal nowhere; what differs between them is which latitude is drawn true. That is the one parameter, and it is the one an affine fit removes.

Mercator against Peters takes apart the most-argued question in cartography and finds both projections doing exactly what they claim. This rung adds a fact about the Peters projection that the argument never reached: it is not a distinct projection from Behrmann’s or Lambert’s, in the strict sense that no measurement of shape can separate them once a reproduction’s anisotropy is unknown.

They are three settings of one number, and the number decides which parallel is drawn at true scale — that is, where on the map the north–south and east–west scales cross. Every member has an areal factor of exactly 1 — which is the property, measured rather than named — and every member destroys shape, and the choice between them is a choice about where the shape damage is least, which is precisely what a standard parallel buys.

A dispute conducted about a projection was, geometrically, a dispute about one parameter of a family whose members differ by a stretch.

What survives an affine fit #

The family is the whole of what an affine fit destroys, and the rest of the library survives it nearly intact.

That is worth measuring rather than assuming, because six parameters fitted to a hundred points could in principle absorb a good deal of anything. It does not: Mollweide against the sinusoidal keeps 1.4 × 10⁻² of residual under the affine fit against 2.2 × 10⁻² under the similarity, and Mercator against the stereographic keeps 1.6 × 10⁻² against 1.7 × 10⁻². The extra freedom buys 40 per cent and 6 per cent, and the pairs stay separable at almost the same extents.

So the loss is exactly one family and one parameter, which is the sharpest form the finding can take: an affine fit is safe for every candidate whose own parameters do not act as a stretch, and catastrophic for the one whose parameter is a stretch.

The warp a georeferencing tool actually fits #

There is a practical version of this that goes considerably further, and it is worth naming because it is what happens to most scanned maps.

A georeferencing tool asked to place a scanned sheet does not fit a similarity or an affine map. It fits a polynomial warp — second order, third order, or a thin-plate spline — because that is what removes the paper’s own stretching and the scanner’s non-linearity, and because it makes the control points line up.

Everything this ladder is about disappears under such a warp. A second-order polynomial absorbs the quadratic difference between two conformal projections — the term two projections that cannot be told apart shows is all that separates them; a third-order one absorbs the term that separates them; a spline with enough control points absorbs any projection difference whatever, because it can bend anywhere.

That is not an argument against georeferencing tools, which are doing what they are asked. It is a statement of what the practice costs: a sheet warped into place has had its projection removed rather than identified, and the warp’s own coefficients are not interpretable as anything cartographic. The identification, if it is wanted, has to happen before the warp and on the graticule.

The one thing an affine fit cannot hide #

There is a limit to what the ambiguity can reach, and it is worth stating because it bounds the damage.

An affine transformation multiplies every area by the same constant — its own determinant. So it can carry an equal-area map onto another equal-area map, and it can never carry an equal-area map onto a map that is not one. The property survives the ambiguity even though the projection does not.

That is why a reader of an affinely-reproduced map is not helpless. The identification cannot say which member of the family it is, and the ranking can still say, with a margin of millions, that it is in the family — which is the fact most readings of a map actually need. Whether the sheet is Gall–Peters or Behrmann changes nothing about whether it can be used to compare two countries’ areas; it can, and every member can.

The same argument runs for conformality: an affine map is not conformal unless it is a similarity, so a conformal map affinely reproduced stops being conformal and the fit notices. The ambiguity is exactly one family wide because the family’s parameter is exactly the nuisance, and no property that the nuisance does not preserve is at risk.

What was computed, and how #

The affine fit is two independent 3 × 3 solves, because the two coordinate equations decouple: x = a₁X + a₂Y + a₃ and y = a₄X + a₅Y + a₆. There is no iteration and no starting guess in either the affine or the similarity case, so the comparison between them is a comparison of two exact solutions rather than of two searches.

The one thing that had to be checked is that the affine fit does not collapse everything. Six parameters fitted to a hundred and twenty points is not close to degenerate, but a fit with enough freedom will absorb any difference, and the check is the pairs that are not affine images: Mollweide against the sinusoidal keeps 1.4 × 10⁻² of residual, which is a per cent and a half of the map’s width and is not going anywhere.

The family’s own identity was verified independently of the fit, by composing the two projections’ formulae symbolically and confirming that the result is (cx, y/c) with c the ratio of the cosines. A measurement that agrees with an algebraic identity to 10⁻¹⁶ is checking the arithmetic; it is the algebra that says why.

What to do about it #

Three things, in increasing order of how much they ask of the person holding the map.

Report the group. A published identification should say which plane transformations were removed, in the same way a published distortion figure should say which criterion was used. Fitted under a similarity and fitted under an affine transformation are different claims and the second is weaker by exactly one family.

Use the ranking’s margin. An affine fit of the cylindrical equal-area family returns three candidates at 10⁻¹⁶ and everything else far behind: a margin of one rather than of millions. That is the ranking saying these are the same map, and it is legible without knowing the algebra in advance.

And fix the scale independently if it can be fixed. The whole ambiguity comes from not knowing the reproduction’s two scales. A sheet with a scale bar, a stated sheet size, or two ground distances marked on it has that information from outside the graticule, and with it the fit can be run under a similarity again — and the family separates at 0.75° like everything else.

That last route is the one a working archive has most often, and it is the reason the ambiguity is a practical nuisance rather than a wall.

Where the model stops #

Only two groups of nuisance transformations are considered, and the choice between them is as consequential as the choice of which projection to trust in the first place. A projective transformation — what a photograph of a flat sheet from an angle produces — has eight parameters and absorbs more; a polynomial warp, which is what a georeferencing tool fits to a scanned map, absorbs a great deal more and is routinely used with no statement at all of what it has thereby made unidentifiable.

The family is exactly affine-equivalent; nothing else in the library is. That is a property of one family and does not generalise to nearly affine pairs, which behave like the ordinary cases and separate at a computable extent.

And the reproduction is assumed uniform across the sheet. Real paper stretches unevenly with humidity, and a scanner’s non-linearity is not an affine map. Both would leave a structured residual under either fit, which is the signature the first rung of this ladder uses to distinguish a wrong candidate from a badly-read map.

The same fact from the area side #

There is a second way to see that the family is one projection, and it uses the quantity the family was invented for.

Every member has an areal factor of exactly 1 everywhere. That is the property they share and it is what the argument about them was ostensibly about — the projection that shows true size takes the phrase apart and finds that equal-area projections preserve area and destroy shape, all of them, in the same way.

So on the quantity they were chosen for, the members are identical by construction. On shape, they differ by exactly the transformation a fit removes. Between those two statements there is nothing left for a measurement to find, which is a sharper way of saying they are one projection with a dial.

The dial is worth something: it decides where the shape damage is least, and a map for a particular band of latitudes should set it there. What it is not is a distinct construction, and a comparison of two members is a comparison of two settings.

The generalisation #

The rule is one that every inverse problem with nuisance parameters has, and it is usually stated as a technical condition rather than as a practical warning:

A parameter of the model that acts on the data in the same way as a nuisance parameter is not estimable, however good the data is.

The cartographic instance is unusually clean because both halves are exact. The nuisance — a reproduction’s anisotropic scaling — is an affine map exactly. The model parameter — the standard parallel of a cylindrical equal-area projection — acts as an anisotropic scaling exactly. So the loss of identifiability is total rather than approximate, and no amount of extent, density or precision recovers it.

The habit that follows is to ask, of any fit that removes a nuisance: which of the model’s own parameters does that nuisance look like? On this site the answer had a name and forty years of argument attached to it.

Who found it, and when #

That the cylindrical equal-area projections form a one-parameter family related by an anisotropic scaling is elementary and appears in every systematic treatment, Snyder’s included. What has not usually been drawn from it is the identifiability consequence, because identifying projections from pictures is a small enough activity that its blind spots are not much discussed.

The dispute the family is famous for ran from 1973, when Arno Peters presented the projection at a press conference, through two decades of cartographic society resolutions. The geometric content — that the projection was Gall’s of 1855 at a different standard parallel, and Lambert’s of 1772 before that — was pointed out immediately and repeatedly. What this rung adds is the strict version: it is not merely that the projections are closely related, it is that no measurement of a reproduced map can distinguish them.

The strictness is what changes the argument’s status. Closely related invites a reply about how close is close enough, and the reply can be argued either way indefinitely; no measurement of a reproduced map distinguishes them is a claim with a test attached, and the test either finds a separating measurement or it does not. Twenty years of resolutions could not have been settled by the first sentence and are settled by the second.

It also relocates the disagreement. If no measurement separates the two, then whatever the argument was about was never geometry, and the participants who insisted it was about something else were right about that much.

Which is a narrower conclusion than either side would have wanted and is the only one the geometry supports.

Where the ladder goes next #

This ladder started with a map that does not say what it is and ends with a map whose reproduction has removed the answer. What it has not done is the thing an identification is usually for: taking the recovered projection and using it to place the map’s own features on the ground, which is georeferencing and brings back the datum question this collection keeps separate — because a projection recovered perfectly still leaves the hundreds of metres a datum moves a coordinate entirely unaddressed.