The nearest equal-area map to an impossible request

The nearest map to an impossible request took a stated distortion field, found the closest achievable one, and measured what was left over. Every number in it is inside the conformal achievable set, because Liouville’s condition is the conformal condition, and the essay recorded that as a shortfall: the equal-area case is a different equation with a whole function of freedom, and the nearest map to an impossible request might be much nearer.

It is not nearer. It is exact, every time, and the price shows up somewhere else entirely.

One equation against two #

The asymmetry is not a matter of degree and it is worth deriving rather than asserting.

A map of a plane region is a pair of functions of two variables, and three conditions are one too many for such a pair. Conformality imposes the Cauchy–Riemann equations: two equations on two unknowns, which is a determined system. Its solutions are rigid — fix the boundary values and the interior is decided — and the consequence for scale fields is Liouville’s condition, that $\log\lambda$ must be harmonic.

Ask for a scale field whose logarithm is not harmonic and there is no conformal map that supplies it. That is a refusal, it is what not every distortion can be asked for established, and the previous rung measured the nearest available thing.

The equal-area condition is one equation: the Jacobian determinant equals a prescribed function. One equation on two unknowns is underdetermined, and an underdetermined system does not refuse. It has a solution space of one whole function’s worth, and the only question is which member of it to take.

And a construction that exhibits it #

The existence is not merely a dimension count. There is a construction, it is elementary, and it produces the map in two nested quadratures with no iteration.

Push the density’s marginal in one coordinate, then its conditional in the other, and the Jacobian is lower triangular with the two diagonal entries multiplying to exactly the requested determinant. That is the triangular map, and this collection built it one field over for a cartogram, which is the same problem with a different vocabulary.

So an areal request is not merely achievable in principle. It is achievable by a recipe, in closed form up to a quadrature, for any positive request whatever.

The price, in a different currency #

Meeting the request costs something and it is not a residual.

A map has four derivatives at a point. Fixing the areal factor fixes one number — their determinant — and leaves three degrees of freedom per point, and the request lands on the angle. Measured on these four requests, the angular deformation the granting adds runs from 15.0° to 24.8° at its worst point.

That is the shortfall’s answer, and it is not the one the shortfall expected. The question is the nearest equal-area map nearer has no answer, because the equal-area map is not near the request — it is the request, and what varies is what it destroys.

What “nearest” even means when nothing is refused #

The shortfall was phrased in terms of distance — is the nearest equal-area map nearer than the nearest conformal one — and the question turns out to be malformed, which is worth spelling out.

A distance needs two things and a metric. In the conformal case the two things are the request and the closest achievable field, and the metric is a root-mean-square difference of logarithms over the region; that is a well-posed and meaningful comparison, and the previous rung made it.

In the equal-area case the achievable set contains the request. The distance is zero, at every strength, for every request tried. So there is no nearest map in any interesting sense — there is a whole infinite family of exactly correct ones, and the interesting question is which of them and what it costs.

That is why the figures here plot a residual against an angle rather than two residuals. Comparing a refusal with a non-refusal on one axis would be comparing a number with a zero, and the zero says nothing about how hard the request was.

Why the freedom is a function and not a parameter #

It is worth being explicit about the size of the solution space, because “underdetermined” understates it.

Given any map meeting the areal request, compose it with any area-preserving map of the region and the composition meets the same request — the determinant of a composition is the product of the determinants, and multiplying by one changes nothing. There are as many area-preserving maps of a region as there are functions on it.

So the answer is not a family with a few parameters to be pinned down by a side condition. It is infinite-dimensional, and any additional requirement short of another equation leaves it infinite-dimensional.

That is every equal-area map is every other one stated as a fact about a solution space rather than about a library, and it is why the equal-area cylindrical projections form a one-parameter family with names on three of its members and nothing distinguishing the rest.

What this says about the previous rung #

It reframes it rather than contradicting it.

The previous rung’s residual is a real quantity and it means what it said: a request for a conformal map with a stated scale field can be refused, and the residual measures the refusal. Nothing here weakens that.

What this rung adds is that the refusal is a property of the condition rather than of the request. The same field, asked of a different condition, is granted in full. So “this distortion cannot be asked for” was always shorthand for “this distortion cannot be asked for conformally”, and the qualification was doing all the work.

The cost rises and the residual does not vanish #

Both curves in the strength sweep are worth reading carefully, because they behave differently and the difference is the point.

The conformal residual starts at 11.4 per cent for the weakest request and rises to 27.7 for the strongest — a factor of 2.4 over a factor of sixteen in strength. It rises slowly because the achievable set has a fixed shape and a stronger request moves further outside it in a direction the set does not extend in.

The angular cost of granting the same requests starts at 4.3° and rises to 84.6° — a factor of twenty over the same range, and it is nearly linear in the strength. It rises fast because the cost is set by the request’s own gradient, and scaling a request scales its gradient in proportion.

So the two currencies do not merely differ in kind. They scale differently, and there is a strength below which the conformal refusal is the more painful of the two and above which it is not. On this family the crossing is somewhere near a strength of 0.2, where the residual is 12.7 per cent and the angular cost is 8.6 degrees — and comparing those two numbers requires a judgement no arithmetic supplies.

The two refusals, side by side #

The conditions refuse in exactly opposite ways, which is a symmetry worth stating.

Conformality determines the map from its boundary values and refuses interior requests. Give it the boundary and it has no freedom left; ask it for something the boundary does not imply and it says no.

Equal area determines nothing from a boundary — its equation has no derivative of the determinant in it, so nothing propagates — and it refuses nothing. It accepts any interior request and leaves the shape entirely open.

So one condition is all constraint and no freedom, the other all freedom and no constraint, and they are the two halves of the impossibility this whole site is built on. Both together force an isometry, which is the theorem; separately, one of them is a straitjacket and the other is a licence.

The refusal, which is the flat request #

Every measurement here has to be capable of returning nothing, and the case that returns it is a uniform areal request.

A constant factor over a fixed region, once the total is matched, is the identity: the map is asked to draw every cell at the size it already is and does. The measured areal residual is $5 \times 10^{-12}$ and the measured angular cost is exactly zero, on the construction that costs 24.8° for a radial request.

So the machinery is capable of reporting no cost and does whenever the request has no shape. Every non-zero number above is a statement about the request rather than about the arithmetic.

What a practitioner takes from this #

Two things, and the second is the practical one.

Do not ask whether a distortion is achievable. Ask whether it is achievable subject to the condition being imposed, because the answer depends entirely on the second half. A field that is impossible conformally may be trivial areally, and vice versa.

And expect the cost to move rather than to vanish. A condition that never refuses has not made the problem easier; it has moved the difficulty into whatever it leaves free. The equal-area condition’s freedom is the anisotropy, and the anisotropy is what a reader of the resulting map sees first.

The link to the cartogram anchor #

This rung is the condition ladder’s half of a result the distortion field is building at the same time, and the two are worth reading together.

A map drawn to a density it was handed starts from the areal request and treats it as a specification, then measures what meeting it costs. This rung starts from the achievability question — the one this ladder has spent eight rungs on — and finds that the areal version of it is empty.

They are the same construction seen from two directions. What the cartogram anchor calls “the construction meets its target exactly” this anchor calls “the achievable set contains the request”, and what the cartogram anchor calls “the shape cost” this anchor calls “the currency the refusal is denominated in”.

That the same two quadratures answer a question in the choosing field and a question in the distortion field is the kind of connection this collection is built to find. Neither ladder could have reached it alone: one had the machinery and no reason to ask about achievability, the other had the question and no construction.

Where the model stops #

The comparison here is between two conditions on a plane region with a square boundary, and the conformal side uses the previous rung’s four-parameter boundary family, which is the lowest order that can be anisotropic and off-centre. A richer boundary family would reduce the conformal residuals somewhat and would not change the sign of the comparison, because no boundary family makes a non-harmonic field harmonic.

What is not treated is the case in between: a condition that is one and a half equations, so to speak — a request for approximate conformality with an areal constraint, which is what most real projection design is. That has a nonempty achievable set and a genuine optimisation inside it, and it is the problem Chebyshev’s criterion solves for one special case and nobody solves in general.

An equation with no derivative of its unknown #

There is one more structural point that explains the whole asymmetry in a sentence, and it is worth isolating.

Liouville’s condition is a second-order partial differential equation: $\nabla^2\log\lambda = 0$. Second-order elliptic equations propagate boundary data into the interior, which is exactly why the boundary decides everything and why an interior request can conflict with it.

The equal-area condition is zeroth order in the determinant: $\det \mathrm{D}T = f$, with no derivative of the determinant anywhere. Nothing propagates. A value at one point constrains no value at any other point, so there is no mechanism by which a request at one place can conflict with a request at another.

That is the whole of it. A condition refuses when it has a propagation mechanism and a request that fights it; a condition with no propagation cannot refuse, because there is nothing for a request to fight.

And it explains why the cost moves rather than vanishing. The determinant is a pointwise constraint on a matrix with four entries, so it pins one number and leaves three, and the three are where the map’s smoothness requirements make themselves felt — the map still has to be a map, continuous and invertible, and that is what turns a pointwise licence into a global shape cost.

Who found it, and when #

Liouville’s theorem on conformal maps is from 1850 and the rigidity it expresses is the reason conformal geometry is a small subject with strong theorems. Moser’s theorem on volume-preserving diffeomorphisms is from 1965 and expresses the opposite: the equal-area condition is flexible enough that its solution set is as large as the space it lives in.

The two are rarely put side by side, because they belong to different literatures — complex analysis and differential topology — and a cartographer meets them as two facts about two kinds of map rather than as two answers to one question. Asking the one question is what makes them comparable, and the answer is that the difference is a count of equations.

What this rung’s gate has to prove #

Two claims, and the second is the one that keeps the first honest.

The first is that every stated areal request is met. The check samples the constructed map’s Jacobian by central difference — not the construction’s own arithmetic — and requires the determinant to match the request to five parts in a thousand over the whole region. All four come in at nine parts in a billion or better, which is the difference operator’s noise rather than a tolerance being scraped.

The second is that a uniform request must cost exactly nothing: zero areal residual and zero angular deformation, because it is the identity. That is the refusal, and without it the first claim would be consistent with a check that reported success for any input.

There is a third, which is the comparison itself: at least one of the four requests must be genuinely refused by the conformal condition, or the essay is contrasting two things that both work. Three of the four are, at residuals between four and seventeen per cent.

What the two refusals say about designing by specification #

The pair sitting side by side is the rung’s real content, and it is worth drawing out what it means for anybody stating a requirement rather than choosing a projection.

A conformal request can be refused outright. The condition is an equation on the requested field, the field either satisfies it or does not, and the answer is available before any construction is attempted. A designer who asks for an impossible conformal distortion is told so, and told by how much.

An equal-area request is never refused. The condition leaves a free function, so there is always a map meeting the areal specification exactly, and the cost is paid in a currency the specification did not mention — angle, which nobody constrained and which the construction is therefore free to spend without limit.

Which makes the second the more dangerous kind of specification to write. A requirement that can fail says so when it has. A requirement that is always satisfiable returns a map, silently, whose unspecified properties may be anything at all — and the specification’s own terms give a reader no reason to look.

The practical rule follows. When a request is granted without complaint, ask what it did not constrain, and measure that. Here the unconstrained quantity is the angular deformation and it is measured; in general it is whatever the condition left free, which is exactly the freedom the counting rungs of this ladder exist to identify.

And the pair is why the ladder needed both essays. One rung measuring only the refusable condition would leave the impression that a specification is a protection; one measuring only the always-satisfiable one would leave the impression that anything can be had. Together they say that whether a request can fail is a property of the condition rather than of the request, and that the conditions differ.

Where the ladder goes next #

Nine rungs of this anchor have asked what a condition determines, what it leaves free, whether it can be met and what the nearest thing is when it cannot. This one closes the shortfall the eighth recorded and opens a plainer question: given that one condition refuses and the other does not, what does a condition that is neither — a bound rather than an equation — leave behind?