Where the control points are

Five rungs of this ladder run the same procedure. A map does not say what it is, so measure some graticule intersections, fit every candidate in a library to them, and rank the candidates by residual. Later rungs make the procedure honest: rung four adds a rejection rule based on the margin between the winner and the runner-up, and rung five shows a small residual has more than one explanation.

All five ask how much to trust the answer. None asks whether there is one.

A fit has five unknowns: the candidate’s own parameter — a standard parallel, a cone constant, a centre latitude — and the four of the plane similarity that carries the map onto the page. The information a control set carries about those five is the normal matrix of the fit, and a normal matrix can be singular. When it is, there is a direction in parameter space along which the residual does not change at all; every value along that direction fits equally well; and no number of points and no measurement precision recovers it.

Six configurations of the same size #

Holding the count is what makes the comparison mean anything. Every practical discussion of control points is about how many to measure; this one is about where, with the how-many removed.

The blind case, and why it is blind #

The clearest failure is not the one a reader would guess.

The equirectangular with standard parallel φ₁ draws x = λ cos φ₁ and y = φ. Changing φ₁ scales the page horizontally and does nothing vertically — an anisotropic scaling, which a similarity fit cannot absorb in general and which is exactly what makes the parameter recoverable in general.

On one parallel, every control point has the same y. The whole configuration lies on a horizontal line, and on a line a horizontal scaling is a similarity. The fit absorbs it exactly.

On one meridian, every point has the same λ, so x is a constant and changing φ₁ moves the whole set sideways. That is a translation, and the fit absorbs that exactly too.

Two standard parallels forty degrees apart, one picture. The smallest eigenvalue comes back at −4.9 × 10⁻²⁰ on the parallel and at exactly zero on the meridian, which is a rank deficiency rather than a small number.

And the count does nothing #

What a fit can learn is decided by where the points are and not by how many there are, which is the opposite of the rule everybody applies. The blind configuration is exactly as blind at five hundred and twelve points as at eight, and the grid is already as good at nine as at a hundred and twenty-one.

The second half of that is worth as much as the first. A configuration that works, works quickly: eight or nine well-spread points determine the parameter, and the next hundred buy accuracy against noise rather than identifiability. So the effort a careful analyst puts into digitising a hundred graticule intersections is buying a fraction of what they think, and the effort they did not put into spreading them over two dimensions is what decided the answer.

A conic trades rather than hides #

The conics behave differently and the difference is the useful part.

On one parallel a conformal conic’s smallest eigenvalue is 1.12 × 10⁻⁴ against 7.44 × 10⁻⁴ on a grid — a factor of 6.6, weakly determined rather than undetermined. On a tight cluster it is 4.6 × 10⁻⁸, sixteen thousand times worse than the grid and by far the worst configuration for it.

So extent is the variable and shape is second. A cluster of any arrangement is nearly blind because every candidate agrees over a small enough region, which is rung two’s result arriving as a property of the normal matrix rather than as a residual comparison.

That decomposition is a sentence a rejection rule needs and cannot currently give. Not the fit is uncertain, but the cone constant and the page’s scale and vertical position are exchangeable in this control set, in this proportion — which tells a reader exactly what to measure next: a second parallel far from the first, which breaks the trade in one step.

The same measurement, on the other side of the site #

The tools this rung uses are already here, on a ladder that has nothing to do with identification.

Two parameter sets, one transformation asks why two national agencies publish different seven-parameter datum transformations for the same pair of datums and both are right. The answer is the same object: the normal matrix of a fit, its smallest eigenvalue, and the direction of the corresponding eigenvector — a combination of a translation and a rotation that the markers cannot separate. That essay computes the ground displacement the weakest direction produces and puts it beside a tolerance.

The two problems are the same problem. In one, a set of markers is fitted to seven parameters; in the other, a set of graticule intersections is fitted to five. In both, the arrangement of the points decides which combinations of parameters can be recovered, and in both the practice is to gather more points and hope.

What differs is that geodesy knows. Network design theory is a mature subject, the conditioning of a datum transformation is routinely computed, and an agency that published a transformation without knowing which parameter combination its markers could not see would be doing something unusual. Projection identification has no equivalent tradition, because it is a young subject whose problems have always been small enough to eyeball.

What a careful identification should print #

Four numbers, and three of them are already computed.

The residual, which every identification prints. The margin over the runner-up, which rung four established as the quantity that distinguishes a real winner from a ceremony. The smallest eigenvalue of the winner’s own normal matrix, which says whether the control set could have distinguished anything. And the weakest direction, which says what to measure next if it could not.

The third and fourth are the additions and they cost one eigendecomposition of a five-by-five matrix. What they buy is the difference between an identification that is right and one that could not have been wrong.

What this adds to the rejection rule #

When the answer is not in the library builds a rejection rule out of the margin between the winner and the runner-up, and finds the margin is 1.6 when the truth is absent and 10¹³ when it is present. That rule answers is the winner the right candidate.

This rung answers a question that comes first: can this control set distinguish anything at all? The two failures look identical in a residual table — a blind configuration produces a tiny residual for every value of the parameter, so the winner’s residual is small and its margin over the runner-up is whatever the arithmetic noise happens to give — and they are completely different situations. One is a successful identification and the other is a coin toss reported to four decimal places.

And the check is free. The normal matrix is already formed by the fit; its smallest eigenvalue is one line more; and it can be computed before any measurement is made, because it depends on where the control points are and not on what was measured there. A careful identification should print it beside the residual.

The advice, in three lines #

Spread the control points in two dimensions. One parallel or one meridian can leave a parameter invisible; a cross of the two cannot. This costs nothing at digitising time and is the single decision that determines whether the identification is possible.

Use the extent, not the count. Nine well-spread points are already as identifiable as a hundred and twenty-one; a hundred points in a cluster are worse than eight across a region. Effort spent on more points inside a small area is buying accuracy against measurement noise and nothing else.

And compute the eigenvalue before measuring anything. It depends only on where the points will be, so it can be checked while the control set is still a plan — which is the same discipline a network’s design has had for fifty years and projection identification has never had.

The measurement this rung could not make #

There is one thing the eigenvalue cannot say and it is worth naming, because it is what a reader most wants.

An eigenvalue is a statement about the design. What a given control set actually recovers is the eigenvalue against the noise — the parameter’s standard error, which is the square root of the corresponding diagonal of the inverted normal matrix, times whatever precision the graticule intersections were measured to. That is one line further on and it is not computed here, because it needs a measurement precision, and a measurement precision is an input this essay declines to invent.

What can be said without one is the ratio. A configuration whose weakest eigenvalue is a thousand times another’s needs a thousand times better measurement to reach the same answer, whatever the absolute precision is. So the table is a statement about relative effort, and it is the one that survives not knowing the instrument.

The two figures answer different halves of the same question. Rung two asks how far apart two projections must be before a fit can separate them; this rung asks whether one projection’s own parameter is separable from the page’s similarity. A control set can pass the first and fail the second, and the failure is silent.

Where the model stops #

Only one intrinsic parameter per candidate. The library’s parameterised candidates have exactly one — a standard parallel, a cone constant or a centre latitude — so the normal matrix is five by five. A real conic has two standard parallels, a real oblique projection has three angles, and the degeneracies among those are richer and are not measured here.

The similarity is the right group and it is a choice. A fit that allowed an affine transformation would absorb more, which rung three measures for the equal-area cylindricals, and one that allowed only a rigid motion would absorb less. Every eigenvalue in this essay is relative to the similarity, and the classification of what is blind would change under either alternative.

The azimuthals are excluded and the exclusion is deliberate. Their centre latitude is a rotation of the sphere rather than a parameter of the map, a gnomonic’s page coordinates run to its horizon, and the normal matrix is then dominated by a handful of enormous entries whose smallest eigenvalue is a statement about floating point rather than about the map. Their numbers are computable and they are not a measurement of anything.

And nothing here has noise in it. The eigenvalues are properties of the design and say what is recoverable in principle. What a given measurement precision recovers is the eigenvalue against the noise, which is the standard error of the parameter — a second calculation, one line further on, that this rung sets up and does not do.

Who found it, and when #

The mathematics is optimal experimental design and it is a century old. The information matrix, its smallest eigenvalue, the directions a design cannot see, and the whole vocabulary of D-optimality and E-optimality were developed in the 1920s and 1930s and are standard in every field that plans measurements — chemistry, agriculture, clinical trials, and, notably, geodesy, where network design theory is a mature subject and where the conditioning of a datum transformation is measured with exactly these tools on this very site.

Projection identification does not use them. The literature on identifying a map’s projection is small, recent and residual-based: fit the candidates, rank them, and reason about the winner. That is a reasonable approach and it inherits an assumption from its own framing — that a control set is a sample, so more of it is better and the arrangement is incidental.

The assumption is false in the strongest possible way. A control set is a design, not a sample, and a design can be blind.

A sample and a design are different objects #

The sentence the essay ends its history on is the whole finding compressed, and it is worth unpacking because the distinction governs far more than projection identification.

A sample is drawn from a population and its value grows with its size. If control points were a sample, then more of them would always help, their arrangement would be a nuisance to be averaged over, and the only question worth asking about a control set would be how many.

A design is chosen, and its value depends on where it is. The points are not drawn from anything; somebody decided where to measure. Two designs of the same size can differ by orders of magnitude in what they can distinguish, and a design can be blind — arranged so that two candidate projections predict identical values at every point in it, whatever the noise level and however many points there are.

Blindness is the property a sampling framing cannot express. A sample can be small, or unlucky, or noisy, and all of those get better with more data. A blind design gets no better at all, because the quantity that separates the candidates is zero at every point measured, and multiplying zero by more points leaves zero.

And the distinction has a diagnostic attached, which is the useful part. Blindness is a property of the design and the candidate set together, computable before any measurement is taken: evaluate each candidate at the proposed points, and look at whether their predictions differ by more than the expected noise. If they do not, the design cannot separate them and no amount of observing will change that.

That check needs nothing from the map. It uses the candidate projections’ own formulae and the intended point positions, both of which are in hand before anybody digitises anything — so the whole question of whether an identification is possible can be settled in advance of the work rather than discovered from an ambiguous residual afterwards.

Which is why the framing matters rather than being a quibble about words. A researcher who believes they hold a sample responds to a weak identification by measuring more points, in the same places, at greater expense, and gets a more precise version of the same non-answer. A researcher who knows they hold a design responds by moving the points — and moving three of them may do what tripling all of them cannot.

One further consequence for reading somebody else’s identification: a published result stating how many control points were used says almost nothing, and one stating where they were says nearly everything. The count is the statistic that gets reported, because it is the one a sampling framing asks for.

A figure showing the arrangement would carry more than either, and takes no more room than the sentence reporting the count.A figure showing the arrangement would carry more than either.

Where the ladder goes next #

The ladder can now say when an identification is possible, when it is weak, what direction the weakness lies in, and — from rung four — whether the answer is in the library at all.

What it still cannot do is choose where to measure. Every configuration in this essay was written down by hand, and the natural next question is the one experimental design exists to answer: given a library, a region and a budget of n points, where should they go? That is a maximisation of the smallest eigenvalue over the placement of n points, it is a standard problem with standard algorithms, and the answer would be the first advice this ladder has ever given about an action rather than about an inference.