A rotation is not absorbed the way a shift is

The datum hides inside the projection’s parameters found that a ninety-nine-metre datum shift is absorbed almost entirely into a plane fit’s four degrees of freedom, and recorded a shortfall in three sentences. The shift it measured is dominated by its translations, and a plane fit has a translation in it. DHDN’s rotations are nearly three arcseconds, a rotation about the Earth’s centre is not a rotation of the page, and a large-sheet fit might therefore see one.

It does not. The rotations are the part the fit hides best, and the reason is better than the expectation was.

Isolating a part, which is one line and one trap #

A seven-parameter transformation has three translations in metres, three rotations in arcseconds and a scale in parts per million, and what each of them does is a separate question from what a fit can see of it. Switching one group on and the others off is a filter over the parameter object, and that is the line the previous rung’s shortfall said was missing.

The trap is the ellipsoid. A datum names a shape as well as a placement — it is the first of the four commitments a coordinate makes — and swapping Bessel’s ellipsoid for WGS84’s produces a displacement field of its own — larger than these rotations and of a different shape. A rotations-alone experiment run across two ellipsoids would be a measurement of the flattening. So the ellipsoid is held fixed whenever a part is isolated, and every number below is the named part acting alone.

What each part leaves behind #

The rotations leave 93 times less than the translations over a sheet a hundred and twenty degrees across.

The obvious objection is that the rotations are smaller, so of course they leave less. That objection is answerable and the answer is not close.

Why, and it is a symmetry #

A rotation of the Earth about its polar axis moves every point by the same change in longitude and by nothing else. Latitude does not move; the shape of the graticule does not move; the whole sphere turns underneath a coordinate system that is defined by that axis.

A map drawn in a normal aspect is built to be indifferent to that. A cylindrical projection’s easting is a function of longitude alone, so a change of longitude is a translation of the page. A conic’s meridians are rays from an apex, so a change of longitude is a rotation of the page about that apex. A polar azimuthal is the same.

A similarity fit has a translation and a rotation in it. So on a cylindrical or a conic map, the entire effect of a polar datum rotation is a transformation the fit removes exactly — and it does:

$$6.6 \times 10^{-12} \text{ of the map's own size, on a sheet 90° across.}$$

That is not “small”. That is the floating-point noise of the fit, and the rotation it is hiding moves the ground 71 metres.

The equatorial axes, which have nowhere to hide #

Apply a rotation of exactly the same magnitude about an equatorial axis instead and the picture changes completely. The conic leaves 7.1 × 10⁻⁷ — a hundred and eight thousand times more — and absorbs only 73 per cent of it.

The reason is the mirror image of the last one. A rotation about an axis through the equator tips the whole graticule: latitudes change, the poles move, and the pattern of displacement on the sheet is a smooth field that is not a translation, not a rotation, not a scale and not any combination of the three.

So the shortfall’s expectation was not wrong about rotations in general. It was wrong about which rotation DHDN has: its polar term is 2.455 arcseconds against 0.202 and 0.045 for the two equatorial ones, so the part of its rotation that the map cannot see is twelve times the part it can.

The same picture, for the rotation that is not there #

The two pictures are the argument. Both are datum shifts of comparable size on the ground — 71 metres and 26 — through the same projection onto the same sheet, fitted by the same four parameters. One leaves a smooth field that no similarity can touch, and the other leaves nothing at all.

This is the same distinction a residual has more than one explanation draws between a wrong datum and a wrong projection: the discriminant is the shape of what is left over rather than its size. Here it separates two parts of the same datum.

The projection decides, not the datum #

The sharpest form of the argument is that the invisible axis is a property of the map rather than of the transformation.

Three regimes, and each of them is predicted by the projection’s own symmetry before any fit is run.

The cylindrical and the conic are invariant under a change of longitude and hide the polar rotation to noise.

The pseudocylindrical ones hide nothing, and this is the row worth stopping on. A sinusoidal or a Mollweide map is also a normal-aspect projection with a polar axis, and it is not invariant under a change of longitude — because its easting carries a factor in latitude, so shifting longitude shears the sheet rather than translating it. Wide at the equator, narrow at the poles: the shift is proportional to the width. Their spread across the three axes is 2.0 and 1.3, which is to say no axis is special at all.

And an azimuthal centred on the equator moves its minimum to the equatorial axis its own centre lies on, at 1.8 × 10⁻⁸ against 6.2 × 10⁻⁷ for the polar one. The rule is the same rule and the answer is different because the axis is.

What this does to identification #

Three consequences, and the first is a warning about a real workflow.

A national map is nearly always drawn on a conic or a transverse cylindrical — which is what a country’s grid is designed to be — and both are in the family that cannot see a polar rotation. So the component of a datum shift that a georeferencing fit is least able to detect is the one most national transformations have most of. The previous rung’s conclusion — that the datum is invisible rather than merely hard to see — is not softened by the rotations; it is reinforced by them.

A pseudocylindrical is the projection to fit if the datum is what is wanted, which nobody would ever say for any other reason. Its lack of a longitude symmetry means every part of the transformation leaves something behind, which makes it a worse map and a better instrument.

And the direction the analysis should run is backwards. Given a projection, its symmetry group can be written down, and every component of a transformation that lies in that group is unrecoverable before a single control point is read. This is the identifiability question asked about the datum rather than about the projection’s parameters, and it has the same shape: what a fit cannot resolve is decided by the model’s own invariances rather than by how many points were measured.

What rung 7 measured, in this light #

So the earlier rung’s conclusion needs no correction, and it needs a sharper statement of what it was measuring. “A datum shift is absorbed by a plane fit” is true, and the honest version is that the translations are absorbed well, the polar rotation is absorbed exactly, and the equatorial rotations — which are the only part with anything to see — are the smallest part of every published transformation in the table. The invisibility is not one effect; it is three, and only one of them could in principle be defeated by a larger sheet.

That also explains a number rung 7 reported and could not account for: the residual grows with sheet size but stays six orders of magnitude below anything measurable. It grows because the translations’ second-order departure from a page translation grows. It stays tiny because the part that would grow fast is 0.2 arcseconds.

The scale, which is the odd row #

The scale parameter behaves unlike either. It is absorbed only 25.8 per cent on the largest sheet — by far the worst of the three — and it leaves 3.5 × 10⁻⁹ behind, which is the smallest residual in the table.

Both facts have the same cause. A scale of 6.7 parts per million moves a point at the centre of the sheet by 0.14 metres, so there is almost nothing to absorb, and what is left is close to the fit’s own noise floor. The absorption percentage is a ratio of two small numbers and is not a useful reading here.

That is worth recording as a caution about the previous rung’s headline statistic as well. An absorption percentage is only meaningful while the thing being absorbed is much larger than the residual floor, and it stops being meaningful smoothly rather than suddenly.

The table is predictable before the fit, and the affine fit closes the last window #

The three regimes are described above as predicted by the projection’s own symmetry, and it is worth writing the prediction out, because it is a table anybody can produce without running anything.

Each projection either does or does not have a one-parameter symmetry under a change of longitude, and if it does, the page transformation that symmetry becomes is either inside a similarity or not:

Every entry in the measured table follows from that column, including the azimuthal’s move of its minimum to an equatorial axis and the pseudocylindricals’ refusal to hide anything. The experiment confirms an argument rather than discovering a pattern, which is the right relation between the two and is why the numbers are quoted to one significant figure and the mechanism at length.

The remaining question is what a larger fit would do, and it has an uncomfortable answer.

An affine fit has six degrees of freedom rather than four: it adds a differential scale between two directions and a shear. An equatorial rotation tips the graticule, and over a bounded sheet the leading effect of a tip is precisely a differential scale plus a shear plus a translation — so an affine fit absorbs an equatorial rotation to first order too, leaving only the second-order part. The one component that a similarity fit could in principle see is the one the extra two parameters are built to remove.

So the choice of fit is a dilemma rather than a refinement, and the two horns are both real. Fit a similarity and the sheet’s own differential shrinkage stays in the residual, at a size comparable with the shape difference between two genuinely different projections. Fit an affine and the shrinkage is removed cleanly and so is the last recoverable trace of the datum. There is no fit that removes the paper and keeps the geodesy, because the two enter the page the same way.

That is worth stating as the practical end of the ladder’s identifiability argument. The question is not how many parameters to fit but which quantities the practitioner is willing to lose, and every additional degree of freedom buys robustness against one nuisance by making one signal unrecoverable. A four-parameter fit is the choice to keep a datum’s equatorial rotations at the price of the paper; a six-parameter fit is the opposite trade; and neither is more careful than the other.

The one escape is the one the table points at, and it is a strange piece of advice. Fit a pseudocylindrical. A map with no longitude symmetry leaves something behind from every part of a transformation, so an analyst who genuinely wants the datum should georeference against a sinusoidal or a Mollweide rendering — a worse map, a better instrument, and a recommendation nobody would arrive at from any consideration of cartography.

Where the model stops #

The fit here is a similarity — four degrees of freedom, uniform scale. An affine fit has six, and the affine ambiguity rung shows what the extra two buy: they absorb a stretch in one direction, which is part of what an equatorial rotation does. Every residual above would fall under an affine fit and the ordering would survive, because a symmetry that is exactly removed cannot be removed more.

The rotations are treated as exact and small, which they are: at three arcseconds the second-order terms are 10⁻¹⁰ of the first and nothing here is sensitive to them.

And the sheets are rectangles in longitude and latitude. A real map covers a country, and where the control points are is the rung that shows a layout can hide a parameter that a symmetric layout resolves. A wedge-shaped control set would change these numbers; it would not change which axis is the invisible one, because that is a property of the projection.

The generalisation #

A fit cannot see anything that lies in the symmetry group of the model it is fitting, and this is a statement to be made before the data is collected rather than after.

Stated that way it covers everything on this ladder. A projection cannot be identified from a small region because every candidate is nearly affine there. A datum cannot be recovered because a similarity absorbs it. A polar rotation cannot be recovered at all on a map that does not care about longitude. Each is the same sentence with a different group in it, and each is decided by the model rather than by the measurement.

The pleasing part is that it also says where to look. The residual that survives is exactly the component orthogonal to the symmetry group, and its shape names the part of the transformation that produced it — which is why the equatorial rotation’s residual field is a picture worth drawing and the polar one’s is a blank sheet.

One further reading of the symmetry table is worth having, because it says what the ladder could still measure. The projections that hide the polar rotation hide it exactly, so the residual they leave for it is a floor rather than a signal and no sheet size changes that. The pseudocylindricals hide nothing, so every component they leave grows with the sheet. A single map fitted in both renderings therefore separates the two classes of component by subtraction, at the cost of one extra fit, and neither rung of this ladder has run that experiment.

Who found it, and when #

That a change of longitude is a symmetry of a normal-aspect projection is the oldest observation in the subject — it is why a central meridian is a free parameter rather than a discovery — and it appears in every treatment of grid design as the statement that the origin of longitude is arbitrary.

The consequence for datum recovery is younger and belongs to photogrammetry and to the georeferencing of historical maps, where the question what can be recovered from a scanned sheet was asked seriously from the 1970s onwards. The general form — a parameter lying in a model’s symmetry group is unidentifiable — is the standard statement of identifiability in estimation theory, and it arrives here as a fact about maps rather than as a fact about statistics.

The experiment is cheap because both renderings use the same control points and the same transformation; only the projection object changes, and the difference between the two residuals is the part of the shift that lies in one projection’s symmetry group and not the other’s. That is a subtraction rather than a fit, and it needs no threshold.

Where the ladder goes next #

Everything on this ladder recovers a static thing from a static sheet. The rotations here are constants; the ground they describe is not, and a velocity needs a frame is the essay in which the same rigid rotation turns up as a choice nobody can measure. What is missing from this ladder is a map whose control points were surveyed at different dates.