Not every distortion can be asked for

This ladder has spent six essays writing projections as conditions and asking what a condition leaves free. Three conditions are one too many; a map can be solved for rather than chosen; one equation leaves a whole function free.

Every one of those starts from a property and asks what maps have it. A cartographer starts from the other end. They know what they want the distortion to look like: none along this line, a little in the middle, more out at the edges, and nothing worse than twice true anywhere. Can they ask for it?

For a conformal map the answer is a single equation, and the equation is the Theorema Egregium wearing different clothes.

The condition #

Write the page’s flat metric as dx² + dy² and the body’s metric, pulled back onto the page, as e^{2u}(dx² + dy²). That form is available exactly when the map is conformal, because a conformal map is one whose pullback is a scalar multiple of the flat metric — which is the same statement as a = b and the same statement as isothermal coordinates existing.

The scalar is the point scale factor: u = −ln k.

The Gaussian curvature of a conformally flat metric has a classical closed form:

$$K = -e^{-2u}\,\Delta u.$$

Demand that the body be a unit sphere — K = 1 — and substitute:

$$\Delta(\ln k) = \frac{1}{k^{2}}.$$

That is Liouville’s equation, on the page, and it is the whole condition. A scale field satisfying it is the scale field of a conformal map of the unit sphere. A scale field that does not satisfy it is not a difficult map to construct; it is not the scale field of any map at all.

The check that costs one line #

The apparatus is worth testing on the request every reader of this site already knows is impossible: no distortion anywhere, k ≡ 1.

The left side is the Laplacian of the logarithm of a constant, which is zero. The right side is 1/1 = 1. The residual is exactly −1, and −1 is the Gaussian curvature of the unit sphere with its sign.

So the impossibility that the whole collection rests on is recoverable from this equation as its simplest special case, and the size of the failure is the size of the curvature. That is not a coincidence; it is the equation being a restatement of the curvature, and it is the reason to trust everything below.

The real projections pass. Mercator’s scale factor on its own page is cosh y, and Δ(ln cosh y) = sech² y = 1/cosh² y — an identity, satisfied to 1.8 × 10⁻⁷ by the numerical Laplacian used here. The stereographic’s 1 + (x² + y²)/4 satisfies it to 1.2 × 10⁻⁷.

And two of the requests fail by amounts of order one:

• a scale rising linearly northwards, k = 1 + y/4, fails by 1.06 relative;
• a scale that rises and then levels off, k = 1 + tanh((x² + y²)/4), fails by 0.41.

Both are entirely reasonable things to want. The second is what a designer would write down if asked to bound the worst distortion on a sheet: rise smoothly from the centre and flatten out so that nothing is worse than twice true. There is no conformal map with that scale field, and there is no near miss — the equation is violated by forty per cent of its own right-hand side.

The request that turns out to be a projection #

One of the requests passes, and it is the most obvious one a designer would write: a scale rising quadratically from the middle of the sheet, held true at the centre.

k = 1 + (x² + y²)/4 satisfies Liouville’s equation exactly, because it is the scale factor of the stereographic projection.

That is the pleasing case and it is worth dwelling on. A designer who reasons “hold the centre true and let the error grow with the square of the distance, which is the gentlest growth that is not linear” has not invented anything; they have re-derived Hipparchus’s projection from a wish about distortion. The set of askable scale fields is small, and the ones that fall out of simple wishes are the ones that already have names.

Solving the equation instead of testing it #

Liouville’s equation is elliptic, and that word carries the rest of the essay. An elliptic equation on a region has solutions determined by their values on the boundary: the interior is not free.

Solved on a 41-square by Newton’s method on the nonlinear five-point stencil, with boundary values taken from a real projection, the interior comes back to 9.9 × 10⁻⁵ of that projection’s own scale factor in 271 sweeps. Change the boundary values and the interior changes everywhere — by up to 51 per cent at the point that moves most — which is the refusal: if the interior were insensitive to the boundary the recovery would be meaningless.

So the freedom in a conformal map’s scale field is exactly one function of one variable, on the edge of the region, and the request was a function of two variables over the whole of it.

The counting makes the point sharper than the analysis does. On an 11-square a request has 81 interior numbers and 40 boundary ones, a ratio of 2. On a 161-square it has 25,281 and 640, a ratio of 40. A finer request is not a more detailed choice; it is a larger fraction of the answer already decided.

Why “nearly” is not available #

The failures above are large and that is the part worth taking seriously, because a reader’'s instinct is that a request which nearly satisfies the equation should have a map that nearly has it.

It does not, and the reason is what elliptic means. An elliptic operator smooths: the solution at a point depends on the boundary data everywhere, weighted by a kernel that decays but never vanishes. So a field that violates the equation by a fixed amount over a patch is not close to a solution — there is no solution nearby to be close to, because the solutions form a set of much smaller dimension and the violation cannot be pushed into a corner.

Stated in terms a designer would use: a request that fails the condition cannot be fixed by adjusting it locally where it is worst. Adjusting it locally changes the Laplacian locally and leaves the rest of the request unsatisfied, and the only way to reach a solution is to abandon the interior request entirely and choose a boundary.

That is why the honest procedure is optimisation rather than repair. Given a request that is not achievable, the question worth asking is which achievable field is nearest to it in some stated norm — and that is a well-posed problem with an answer, which is what Chebyshev’'s criterion provides for one particular choice of norm and region.

What this says about the equal-area case #

The companion rung found the opposite result for the other condition: equal-area leaves a whole function free, because ab = 1 is one equation on two unknowns and the family of area-preserving maps of the plane is infinite-dimensional.

Both statements are true and the difference between them is instructive. Equal-area is one scalar equation on a map, and it constrains the map without pinning the scale factors individually — a may be anything as long as b is its reciprocal. Conformality is also one scalar equation, a = b, but the shared value is then forced by the geometry, because a conformally flat metric’s curvature is determined by it.

So the two great conditions of this subject differ in a way that is invisible from their statements and total in their consequences:

• ask for equal area, and there is an infinite-dimensional family to choose from, which is why there are dozens of named equal-area projections and no way to enumerate them;
• ask for conformality, and the scale field is determined by the boundary, which is why the conformal projections form a rigid family and why solving for the optimal one over a region has a unique answer.

What was computed, and how #

The residual is a five-point Laplacian of ln k at a stated step against the closed form 1/k², evaluated at five points on the page rather than one, so a field that happens to satisfy the equation at the origin is not recorded as satisfying it.

The step is 10⁻³ of a page unit. It is large enough that the second difference is not dominated by cancellation and small enough that its own truncation is below the 10⁻⁷ the real projections come in at — and the check that this is the right regime is that the two real fields agree at the same order rather than at different ones.

The solver is Newton’s method applied pointwise to the nonlinear stencil, with the exact derivative of the right-hand side, over-relaxed by a factor of 1.9. That factor is not cosmetic: a plain Gauss–Seidel sweep on a 41-square contracts by 1 − O(1/n²) per sweep, so the first version of this solve ran four hundred sweeps and stopped three per cent short of the answer — a number that would have been reported as the equation’s own inaccuracy rather than as an unfinished iteration.

The pair of checks is deliberate. Boundary data from a real projection must reproduce that projection’s interior, and different boundary data must produce a different interior. Without the second, a solver that ignored its boundary entirely and always returned the same field would pass the first.

Where the model stops #

Conformal only. The equation above is for maps with a = b. For a general map the compatibility condition is the full Gauss equation relating the two scale fields and their derivatives, and it is a system rather than a single scalar equation. What survives is the shape of the conclusion: a distortion field is not freely prescribable, because the body’s curvature constrains it.

A unit sphere. For a body of curvature K(x, y) the right-hand side becomes K e^{2u}, so a variable-curvature body — which every real body is — gives an equation with a variable coefficient and the same character.

A square patch, not a sphere. The solve is on a rectangle of the page with Dirichlet data, which is the local problem. Asking for a scale field on the whole sphere adds a global constraint that a plane cannot satisfy at all, which is why every world map has a cut or a singularity.

Nothing here is an algorithm for designing maps. A designer given a boundary condition and a solver still has to choose the boundary condition, and the relation between what they want and what boundary produces it is not addressed.

The generalisation #

The result restates the collection’s central fact in the vocabulary of design rather than of measurement.

The usual statement is a prohibition: no map is faithful. The statement here is a description of what is available: the achievable distortion fields are the solutions of a second-order elliptic equation, parameterised by their boundary values. That is a small set — a function of one variable where the request was a function of two — and every named conformal projection is a point in it.

The prohibition is the special case where the request is the constant field, and the general case says something the prohibition does not: that near misses are not available either. A designer cannot ask for almost no distortion, or for distortion shaped like a chosen curve, and get a map that nearly does it. They can choose a boundary and take what comes.

Which is why the practical route this collection uses is the one it uses: choose an objective and solve for the optimum. Optimising over the achievable set works; prescribing a member of it does not, because the set is much thinner than the description of one of its members.

Who found it, and when #

Liouville’s equation appears in Joseph Liouville’s work of the 1840s, in his notes appended to Monge’s Application de l’analyse à la géométrie, and the relation between a conformally flat metric’s curvature and the Laplacian of its conformal factor is standard nineteenth-century surface theory.

Its status as a constraint on cartographic design is not standard, largely because design in this subject has historically proceeded by construction rather than by specification: a projection is proposed as a formula and its distortion is then measured, so the question of whether an arbitrary distortion field is achievable does not arise.

Where it does arise is in the minimum-distortion literature. Chebyshev’s criterion of 1856 and Grave’s proof of it in 1911 are statements about the optimum over the achievable set, and the achievable set is exactly the solution set of the equation above — which is why the criterion takes the form of a boundary condition rather than of a formula.

Why the constraint is invisible to the usual design method #

There is a reason a condition this fundamental is absent from the cartographic literature, and it is not that anybody overlooked it. The standard method cannot encounter it.

A projection designed by construction starts from a formula — a conic with two standard parallels, a pseudocylindrical with a chosen meridian shape, a polynomial in the complex plane — and then measures what distortion the formula produces. Everything reached that way is achievable by definition, because a map was written down first and the field was read off it. The equation is satisfied automatically and never appears.

The same is true of the minimum-distortion tradition. An optimisation over a parameterised family searches inside the family, and the family is a set of maps. The constraint is enforced by the parameterisation itself, invisibly, and the optimiser never proposes a field that no map realises because it never proposes a field at all.

The constraint becomes visible exactly when the design variable changes. A method that parameterises the distortion field — a spline surface of scale factors, a fitted field of Tissot axes, a network trained to output a deformation at each point — is working in a space that is much larger than the achievable set, and almost every point in it corresponds to no map whatever. Such a method must impose the equation as a hard constraint or it will converge happily to a specification that cannot be built, and it will give no sign: the fitted field will look smooth, plausible and locally reasonable everywhere.

That is a live failure mode rather than a hypothetical, because parameterising the output field is the natural thing to do when the design goal is stated as this region should have this much distortion and that region that much — which is how a user states it, and which is the whole reason the specification-first approach is attractive.

And the failure is not detectable by sampling. Checking that the requested field is achievable at a hundred places says nothing, because the condition is a second-order differential relation and a field can satisfy it pointwise-looking everywhere while failing it as a derivative. The test is the one this essay gives — apply the operator, compare with the curvature — and it costs a line precisely because it is the whole condition rather than a sample of it.

So the honest statement of this rung’s practical reach is narrow and real: it changes nothing for anybody designing projections the way projections have been designed for four centuries, and it is the first thing anybody designing them the new way needs to know.

Where the ladder goes next #

Seven rungs have written projections as conditions, found where conditions conflict, solved for maps rather than choosing them, found a map with no formula, found that one equation leaves a function free, and now found what the other equation does not leave free.

The natural next question is the same one on a body that is not a sphere, where the right-hand side carries the body’s own curvature and the achievable fields are correspondingly stranger — and this collection already has the bodies to ask it about.