The nearest map to an impossible request

The previous rung put a question this ladder had spent six essays approaching from the other side. Every earlier rung writes a projection as a condition and asks how much freedom the condition leaves. Rung seven reversed it: a cartographer knows what distortion they want, so can they ask for it?

For a conformal map the answer is a single equation — Liouville’s, which is the Theorema Egregium in disguise — and most fields anybody would want do not satisfy it.

What it did not ask is what happens when one is asked for anyway. That question has a shape, and the shape is the same one every least-squares problem has: the achievable fields are a set, the request is a point off it, and the interesting objects are the nearest point of the set and what is left over.

What the set of achievable fields is #

Rung seven’s own machinery says what the set looks like, and it is smaller than it appears.

Liouville’s equation is elliptic, so a solution on a patch is fixed by its values on the boundary alone. That is the size of the freedom: one function of one variable, where the request is a function of two. Rung seven measured it — 1,521 interior values decided by 160 boundary ones — and left it as a statement about dimension.

Here it becomes the search space. Finding the nearest achievable field means searching over boundary values, not over interior ones, and the interior comes out of the solver. Four coefficients of a low-order boundary form turn out to be enough: higher orders buy under two per cent of the residual, which is the check that what is left is the request’s impossibility rather than the parameterisation’s poverty.

The three impossible requests, and why each is #

The three requests are not arbitrary; each is something a cartographer would actually write down.

A constant scale, not equal to one. A magnified isometry — the same scale everywhere, just not unity. It fails by a relative residual of exactly 1.00, because Liouville’s equation for a constant field reduces to the curvature itself, and the residual is the sphere’s curvature in the units the equation is written in. That is the same 1 that rung seven found for the isometry, and it is the cleanest number in this essay.

True north–south, stretched east–west. The specification of a cylindrical projection restricted to a patch. It fails by 599, which is the largest residual of the three and produces the smallest ungranted part — 3.7 × 10⁻² — because failing an equation badly at a point is not the same as being far from the achievable set. The two quantities measure different things and are not even ordered together.

Scale falling with distance from the centre. What a cartographer draws when asked to sketch a good regional map: true in the middle, gently worse outward. It fails by 2.56 and leaves the largest residual, 1.8 × 10⁻¹, because it asks for a shape the equation cannot make rather than a value it cannot reach.

That the ranking by Liouville residual and the ranking by ungranted part disagree completely is worth stating on its own. A condition’s residual at a point says whether a field is achievable; only the fit says how far from achievable it is, and the essay that measured the first could not have predicted the second.

What “nearest” has to mean #

A least-squares problem needs a metric, and the choice made here is not the obvious one.

The misfit is measured in the logarithm of the scale factor rather than in the scale factor itself. That is deliberate and it matters: a scale of 2 and a scale of ½ are equally wrong in opposite directions, and only the logarithm treats them so. Measuring in the raw factor would weight a doubling more heavily than a halving and would make the fitted map systematically too small — which is the same reasoning behind the Kavrayskiy criterion the choosing ladder uses, where the squared logarithms of the two principal scales are what is summed.

The second choice is the region the misfit is summed over: the whole patch, unweighted. That is an opinion in exactly the sense distortion over a region established — a different weighting gives a different nearest map — and it is stated rather than defended. What does not depend on it is the floor below: no weighting makes a constant scale achievable.

Four requests, and one that is granted #

The first row is the refusal and it has to be there. A field a conformal map can already produce is produced, to 5.1 × 10⁻⁴ — which is the solver’s own discretisation on a 31-square grid and not a disagreement. So the search is capable of returning zero, and the other three numbers are measurements rather than artefacts of a fit that never converges.

The other three fail Liouville’s equation in the first place, by relative residuals of 1.00, 2.56 and 599, and leave between seventy and three hundred and fifty times the floor ungranted.

The residual is a picture, not a number #

An RMS says how much and not where, and where is the part that could not have been guessed.

The hero figure is the residual of the radial request, and its structure is the finding: the request is over-supplied in a ring and under-supplied at the centre and the corners. That is not what a smoothing would do, and it is not what a uniform shortfall would look like. It is the shape of the difference between a function somebody wanted and the nearest solution of an elliptic equation, and it says that the impossibility of that request is concentrated, not spread.

Along one line the two curves cross rather than one lying under the other. A projection that granted the request in the middle and fell short at the edges would be a compromise; this is not a compromise, it is a different function that happens to be as close as the equation allows.

The equation is doing the work, not the fit #

It is worth separating two things that a residual can be blamed on.

The parameterisation. Four boundary coefficients is a small family, and a residual could be measuring that smallness rather than any impossibility. Raising the order to six and then to ten moves the residual by under two per cent, which says the boundary form is not the constraint — and the achievable case coming back at the solver’s own floor says the same thing from the other direction, because a poor parameterisation would fail there too.

The equation. What remains is the elliptic constraint itself: the interior is determined by the boundary, so once the boundary is chosen there is nothing left to adjust. That is what makes the set small enough for a request to be off it. A condition that constrained less — one leaving a function of two variables free — would have an achievable set large enough to contain almost anything, and the whole question would not arise.

Which is the freedom measurement rung seven made turned into a consequence: the reason a distortion cannot be asked for is exactly that the condition determines the interior from the boundary, and the reason the nearest map is findable at all is that the same fact makes the search space one-dimensional.

Asking for less does not help #

The obvious next move is to ask for less: if a request is impossible, a gentler version should be nearly possible. That expectation is wrong here and the reason is the whole collection’s subject.

A factor of twenty in the request buys a factor of two in the residual, and the residual does not head for zero. The reason is that the gentle limit of scale falling with radius is a constant scale — and a constant scale is an isometry, which is the most impossible request there is.

So the floor of that curve is not a numerical artefact and it is not the parameterisation. It is the curvature of the sphere, appearing as the residual of a least-squares problem, and the measurement is that a request cannot be made achievable by moderating it because moderating it moves it towards the one thing that is certainly forbidden.

What the solver refuses, and why it is right to #

One practical thing that had to be handled and turned out to be a finding.

The natural way to start the search is from the request’s own boundary values. For the achievable case that is exact at the first step. For the others it sometimes fails outright: the solver diverges to NaN rather than returning a wrong answer.

That is the equation refusing. A boundary near a constant of one is a boundary near the isometry, and the isometry has no solution at all — so the Newton iteration has nothing to converge to and says so. The fix is to blend the start towards a boundary the equation certainly can carry and proceed from there, and the diagnostic value is worth keeping: the solver’s divergence is a detector for the flat case, sharper than any residual, because a residual is a number and a divergence is a category.

What a cartographer should take from it #

Three readings, and the third is the one that changes practice.

A request is a specification, and most specifications are unsatisfiable. Rung seven established that; this one prices it. The relevant number is not whether a field is achievable — almost none is — but how much of it can be had, and that is computable.

The residual is where the specification has to be renegotiated. A specification is usually written as a whole and refused as a whole, and this makes it possible to refuse it in parts. A single RMS invites the wrong response, which is to accept the nearest map and move on. The residual’s shape says which part of the request the equation objected to, and a cartographer who sees that the objection is concentrated in the corners can drop the corners from the specification and get the rest.

And a conformal map is the wrong family for most requests. Every number here is for the conformal set, because Liouville’s is the conformal condition. Solving for the map instead of choosing it fits a general map by least squares over a much larger family and gets closer to any given target — and a map with no formula is what that produces. The right reading of a large residual here is often stop asking for conformality, which is a decision the residual can prompt and cannot make. That is the honest limit of the whole method: it prices a request against one family and says nothing about whether the family was the right one to ask.

Where this leaves the ladder #

Eight rungs in, the anchor’s arc is now complete in one direction and has a clear next step in the other.

The ladder began by writing a projection as a condition rather than a formula and asking what a condition determines. It found that three conditions are one too many, that some conditions leave no inverse, that a condition can be solved rather than chosen, that the solution may have no formula at all, that one condition leaves a whole function free, and that not every distortion can be asked for.

This rung closes the loop: given a request the condition cannot satisfy, there is a nearest map, it is computable, and what it leaves behind is a picture rather than a number.

What is left open is the other family. Every measurement here is inside the conformal set, because Liouville’s equation is the conformal condition — and the equal-area set is a different set, defined by a different equation, with its own achievable fields and its own nearest maps. Recorded as a shortfall: the same search over equal-area fields, where the freedom is a whole function of one variable rather than a boundary and the nearest map to an impossible request may be much nearer.

What a designer does with a residual that has a shape #

The residual being a picture rather than a number is the finding with the most immediate use, and it is worth saying what the use is.

A scalar residual answers how far, which is not an actionable question. Told that a specification is 12 per cent ungranted, a designer can accept it, abandon it, or moderate it — and the essay’s own measurement shows that moderating it barely helps, since the floor is the sphere’s curvature and not the ambition of the ask.

A shaped residual answers where, and that is actionable. The unmet part of the request is concentrated rather than spread, so a designer can see which region of the specification the geometry objected to. That converts an impossible requirement into a negotiation about a particular place: this corner of the map cannot have what was asked, and the rest can.

Which is the form a real specification arrives in. Nobody asks for a scale field as an analytic function; they ask for low distortion over the populated middle, tolerable distortion over the periphery, and do not much mind about the corners. A residual concentrated in the corners is a granted request; the same residual concentrated in the middle is a refused one; and the scalar figure is identical in both cases.

It is also the half that survives a change to the criterion. A distance depends on which norm was chosen to measure it, so two studies using different norms report different numbers for the same request; the region where the residual concentrates is much less sensitive to that choice, because it is decided by where the curvature objects rather than by how the objection is totalled.

So the right output of such a solve is two things, and only one of them is currently reported. The distance to the achievable set says whether the request was possible. The residual field says which part of it was not, and it is the half a designer can act on.

What this rung establishes #

The achievable fields are a set fixed by a boundary, so the nearest one to a request is found by searching over one function of one variable rather than two — which is rung seven’s freedom measurement used as a search space rather than quoted as a dimension.

A request the equation can grant is granted to 5.1 × 10⁻⁴, the solver’s own floor, and three that it cannot leave 3.7 × 10⁻² to 1.8 × 10⁻¹ — seventy to three hundred and fifty times that floor.

The residual has a shape, concentrated rather than spread, and the shape is what says which part of a specification the curvature objected to.

A condition’s residual and the distance to the achievable set are different quantities, and they rank the three requests in opposite orders: the one that fails Liouville’s equation by 599 leaves the least ungranted, and the one that fails by 2.56 leaves the most.

And the residual has a floor that moderating the request does not reach. A factor of twenty in the ask moves it by a factor of two, because the gentle limit of any scale request is a constant scale, a constant scale is an isometry, and the floor of the whole problem is the sphere’s own curvature. That is the ladder’s own founding statement arriving as the output of an optimisation rather than as its premise, which is the form this collection prefers: the curvature was not put into the search, and it is what the search returns when asked for something it cannot have.